Modern real analysis
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Jacek Hejduk – University of Łódź, Faculty of Mathematics and Computer Science Department of Real Functions, Banacha 22, PL-90-238 Łódź Ryszard J. Pawlak – University of Łódź, Faculty of Mathematics and Computer Science Department of Methodology Teaching Mathematics, Banacha 22, PL-90-238 Łódź Stanisław Kowalczyk, Małgorzata Turowska – Pomeranian University in Słupsk Institute of Mathematics, Arciszewskiego 22b, PL-76-200 Słupsk REVIEWERS Marek Balcerzak, Artur Bartoszewicz, Szymon Głąb, Jacek Hejduk, Grażyna Horbaczewska, Jacek Jędrzejewski, Stanisław Kowalczyk, Tomasz Natkaniec, Jurij Povstenko, Franciszek Prus-Wiśniowski Marian Przemski, Elżbieta Wagner-Bojakowska, Wojciech Wojdowski INITIATING EDITOR Damian Rusek TYPESETTING Stanisław Kowalczyk, Małgorzata Turowska PHOTO Małgorzata Terepeta COVER DESIGN Katarzyna Turkowska Cover image: © Depositphotos.com/Glass cubes Printed directly from camera-ready materials provided to the Łódź University Press This publication has been financed by the Faculty of Mathematics and Computer Science © Copyright by Authors, Łódź 2015 © Copyright for this edition by Uniwersytet Łódzki, Łódź 2015 Published by Łódź University Press First Edition W.07154.15.0.K Printing sheets 14,125 ISBN 978-83-7969-663-5 e-ISBN 978-83-7969-955-1 Łódź University Press 90-131 Łódź, 8 Lindleya St www.wydawnictwo.uni.lodz.pl e-mail: [email protected] tel. (42) 665 58 63

Dedicated to Professors

Roman Ger, Jacek Jędrzejewski, Zygfryd Kominek

Contents

Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface Jacek Hejduk, Stanisław Kowalczyk, Ryszard J. Pawlak and Małgorzata Turowska

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Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Quasicontinuous functions with small set of discontinuity points Ján Borsík Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Convolution operators on some spaces of functions and distributions in the theory of circuits Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Continuity connected with ψ-density Małgorzata Filipczak and Małgorzata Terepeta Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 On equivalence of topological and restrictional continuity Katarzyna Flak and Jacek Hejduk Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 On fields inspired with the polar HSV − RGB theory of Colour Ján Haluška

6

Contents

Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Generalized (topological) metric space. From nowhere density to infinite games Ewa Korczak-Kubiak, Anna Loranty and Ryszard J. Pawlak Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Path continuity connected with density and porosity Stanisław Kowalczyk and Małgorzata Turowska Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Topological similarity of functions Ivan Kupka Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 On the Darboux property of derivative multifunction Graz˙ yna Kwieci´nska Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 On extensions of quasi-continuous functions Oleksandr V. Maslyuchenko and Vasyl V. Nesterenko Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Weak convergence with respect to category Władysław Wilczy´nski Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 New properties of the families of convergent and divergent permutations - Part I Roman Wituła, Edyta Hetmaniok and Damian Słota Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 New properties of the families of convergent and divergent permutations - Part II Roman Wituła

Chapter 1

Preface

JACEK HEJDUK, STANISŁAW KOWALCZYK, RYSZARD J. PAWLAK, MAŁGORZATA TUROWSKA

The first conference on Real Functions Theory had been organized by Mathematical Institute of the Slovak Academy of Sciences in Bratislava since 1971. Since 1972 conferences were organized by other institutions every second year and since 2005 a Polish group of mathematicians joint to organize conferences every second year. So since 2004 International Summer Conferences on Real Functions Theory held every year. The latter conference is XXIX, one in the series that consists of: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

1971 — Modra - Harmónia ˇ 1972 — Cingov 1974 — L’ubochˇna 1976 — Modra - Harmónia 1978 — Modra - Harmónia 1980 — Trnava 1982 — Dolné Orešany 1984 — Kamenný Mlyn 1986 — Richˇnava 1988 — Dubník 1990 — Dubník 1992 — Dubník 1994 — Liptovský Ján 1996 — Liptovský Ján

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(15) (16) (17) (18)

Jacek Hejduk, Stanisław Kowalczyk, Ryszard J. Pawlak, Małgorzata Turowska

1998 — Liptovský Ján 2000 — Liptovský Ján 2002 — Stará Lesná 2004 — Stará Lesná

First 18 conferences were organized by Slovak Academy of Sciences. Since 2005 in odd years the International Summer Conferences on Real Functions Theory are organized by Pomeranian Academy in Słupsk, University of Łód´z, Łód´z Technical University and University of Computer Sciences and Skills, and in even years the International Summer Conferences on Real Functions Theory are organized by Slovak Academy of Sciences. (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29)

2005 — Rowy (Poland) 2006 — Liptovský Ján (Slovakia) 2007 — Niedzica (Poland) 2008 — Stará Lesná (Slovakia) 2009 — Niedzica (Poland) 2010 — Liptovský Ján (Slovakia) 2011 — Złoty Potok (Poland) 2012 — Stará Lesná (Slovakia) 2013 — Niedzica (Poland) 2014 — Stará Lesná (Slovakia) 2015 — Niedzica (Poland)

Thus in 2015 we celebrate 10th anniversary since Pomeranian Academy in Słupsk, University of Łód´z and Łód´z University of Technology joined Slovak Academy of Science and started organizing the International Summer Conferences on Real Functions Theory every odd year. As far, six conferences organized by Polish institutions were held, four in Niedzica in Pieniny Mountains, one in Rowy near Baltic Sea and one in Złoty Potok in Central Poland. International Summer Conferences on Real Functions Theory has a wide group of regular participants. Three of them, Professor Roman Ger, Professor Jacek J˛edrzejewski and Professor Zygfryd Kominek, celebrate this year seventieth birthday anniversary. Professor Roman Ger was born at July 30, 1945. He received his master’s degree in 1968 at a branch of Jagiellonian University in Katowice. In 1971 he received Ph. D. degree and habilitation in 1976, both from Silesian University in Katowice. He became a full professor in 1990. Professor Roman Ger started research under the supervising of Professor Marek Kuczma, the founder of Polish school of functional equations. From 1976 till now he works at Silesian University. From 2005 to 2008 he was Deputy Director and from 2008 to 2012

1. Preface

9

he was Director of the Mathematical Institute. Between 1990 and 1992 he was a Vice Chancellor of Silesian University. Simultaneously, he worked in Academy of Jan Długosz in Cz˛estochowa in years 1981-1987, 1991-2010 and in Silesian University of Technology in Gliwice in 1988-1991. He was a visiting professor at the University of Waterloo, Waterloo, Ontario, Kanada, 1979; University of Central Florida, Orlando, Floryda, USA, 1987-88; Karl-Franzens-Universität, Graz, Austria, 1994; Universitaät Bern, Berno, Szwajcaria, 1995. In years 1984-2000 he led, founded by Professor Marek Kuczma, the seminar on Functional Equations. In 1981 he founded and leads till now the seminar on Functional Equations and Inequalities. His scientific output contains more than 110 scientific papers, mainly devoted to functional equations and inequalities. He gave more than 50 lectures among the other Universities in USA, Canada, Germany, Italy, Austria, Switzerland, Spain, Greece, Hungary, Czech Republic, Denmark, Israel, China Republic and Venezuela. He was a supervisor of 15 Ph.D. students and has served as a referee for 26 Ph.D. thesis, 8 habilitation dissertations and 18 applications for professor nomination. Professor Jacek J˛edrzejewski was born at August 6, 1945. He received his master’s degree in 1969, Ph. D. degree in 1974 and habilitation degree in 1984, all of them from University of Łód´z. He worked at the University of Łód´z in years 1969-1984. He was a Senior Lecturer in Rivers State University of Science and Technology, Port Harcourt, Nigeria, 1984-1988. After his return to Poland he worked at Higher Pedagogical School in Bydgoszcz, Pomeranian Academy in Słupsk 1989-2005, University of Computer Sciences and Skills 1999-2011 and Academy of Jan Długosz in Cz˛estochowa 2005-2015. From 1988 to 1993 he was a Deputy Director and from 1993 to 1995 he was the Director of the Mathematical Institute at Higher Pedagogical School in Bydgoszcz. From 1989 to 1990 he was a Vice Dean and from 1990 to 1993 he was a Vice Chancellor at Higher Pedagogical School in Bydgoszcz. He also was the Director of the Mathematical Institute at Pomeranian Academy in Słupsk 1999-2005.

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Jacek Hejduk, Stanisław Kowalczyk, Ryszard J. Pawlak, Małgorzata Turowska

His scientific output contains more than 40 papers, mainly devoted to function theory and topology. He was the supervisor of 3 Ph.D. students and has served as a referee for 4 Ph.D. thesis. In 1997 Professor Jacek J˛edrzejewski organized the first International Conference on Real Functions Theory in Ustka. Next conferences took place in 2001 in Łeba and in 2003 in Rowy. In 2005 these conferences evolved into International Summer Conferences on Real Functions Theory. Professor J˛edrzejewski was the president of all International Summer Conferences on Real Functions Theory organized by Polish Universities. Professor Zygfryd Kominek was born at November 8, 1945. He received his master’s degree in 1968 at a branch of Jagiellonian University in Katowice. In 1974 he received Ph. D. degree from Silesian University in Katowice and in 1991 he received habilitation degree from Warsaw University of Technology. He became a full professor in 2008. From 1976 till now he works at Silesian University. From 1992 to 1996 and from 2011 till now he was a Deputy Director and from 1996 to 2002 he was the Director of Mathematical Institute. Simultaneously, he worked in Academy of Jan Długosz in Cz˛estochowa 1992-1998, 2004-2006, in Łód´z University of Technology branch in Bielsko-Biała 1991-2001, in The University of Bielsko-Biała 2001-2003 and in the Katowice Institute of Information Technologies 2006-2011. His scientific output contains more than 70 papers, mainly devoted to conditional functional equations, convex functions and systems of functional equations. He was the supervisor of 5 Ph.D. students and has served as a referee for 16 Ph.D. thesis. The presented monograph Modern Real Analysis is dedicated to the mentioned anniversaries. It contains several chapters, written by frequent participants of International Summer Conferences on Real Functions Theory, where their actual research topics are presented.

Chapter 2

Quasicontinuous functions with small set of discontinuity points

JÁN BORSÍK

2010 Mathematics Subject Classification: 54C30, 54C08. Key words and phrases: quasicontinuous functions, cliquish functions, points of continuity, set of first category, set of measure zero.

2.1 Introduction The definition of quasicontinuity for real functions of real variable was given in [34] by S. Kempisty. Nevertheless, R. Baire in his work [1] has shown that a function of two variables continuous at each variable is quasicontinuous. An independent definition was given by W. W. Bledsoe [2] in 1952 under the name neighborly function. S. Marcus in [49] proved that the notions of neighborly and quasicontinuous functions are equivalent and he developed further properties of quasicontinuous functions. He showed that quasicontinuous functions need not be (Lebesgue) measurable and for each countable ordinal α there is a quasicontinuous function in the Baire class α + 1 which does not belong to Baire class α. N. Levine in [44] introduced the notion of semi-continuous function as a function for which the inverse image of every open set is a semi-open set (a set A is semi-open if A is a subset of the closure of the interior of A). A. Neubrunnová in her paper [53] has shown that the notions of quasicontinuity and semicontinuity in the sense of Levine are equivalent. Z. Grande in [33] has shown

12

Ján Borsík

that a function f is quasicontinuous if and only if the graph of the function f restricted to the set of all continuity points of f is dense in the graph of f . A fundamental result concerning continuity points is due to N. Levine [44] for functions with values in a second countable space (and for functions with values in a metric space [53]) is that the set of discontinuity points of a quasicontinuous function is small. Theorem 2.1. Let X be a topologocal space and let Y be a second countable space ([44]) or let Y be a metric space ([53]). If f : X → Y is a quasicontinuous function then the set of discontinuity points is of first category. So, quasicontinuous functions have the Baire property. On the other hand, if X = R2 [19] or if X is a Baire pseudometrizable space space without isolated points (or X is a Baire resolvable perfectly normal locally connected space) [5] or X is a hereditarily separable perfectly normal Fréchet-Urysohn space [50], then for each Fσ -set A of first category there is a quasicontinuous function f : X → R such that A is the set of all discontinuity points of this function. Points of quasicontinuity were characterized in [45]. Quasicontinuous functions were investigated very intensively. We recommend a survey [52] published in 1988 with more than 120 references.

2.2 Basic definitions Let R, Q and N be the set of all real, rational and positive integer numbers, respectively. For a set A ⊂ R denote by Int A and Cl A the interior and the closure of A, respectively. Recall that a function f : X → Y (X and Y are topological spaces) is said to be quasicontinuous at a point x if for each neighbourhood U of x and each neighbourhood V of f (x) there is an open nonempty set G ⊂ U such that f (G) ⊂ V [34]. H. P. Thielman introduced cliquish functions: A function f : X → Y (X is a topological space and (Y, d) is a metric space) is said to be cliquish at a point x ∈ X if for each neighbourhood U of x and each ε > 0 there is an open nonempty set G ⊂ U such that d( f (y), f (z)) < ε for each y, z ∈ G [63]. Denote by C( f ), D( f ), Q( f ) and K( f ) the set of all continuity, discontinuity, quasicontinuity and cliquishness points of f , respectively. A function f is quasicontinuous (cliquish) if Q( f ) = X (K( f ) = X). Further, denote by C (X,Y ), Q(X,Y ) and K (X,Y ) (or briefly C , Q and K ) the family of all

2. Quasicontinuous functions with small set of discontinuity points

13

continuous, quasicontinuous and cliquish functions. Evidently C( f ) ⊂ Q( f ) ⊂ K( f ) and C ⊂ Q ⊂ K (if Y is a metric space). The set D( f ) for cliquish functions is of first category and if X is a Baire space then f is cliquish if and only if the set C( f ) is dense in X. The triplet (C( f ), Q( f ), K( f )) is characterized in [5], [17], [16], [18]. The notion of strongly quasicontinuous function was used by Z. Grande for s.q.c. functions [26]. In this paper, we will use the notion of strong quasicontinuity for any classes functions between continuous and quasicontinuous functions with the set of discontinuity points of measure zero and all such classes of functions will be called stronly quasicontinuous.

2.3 Quasicontinuous functions with sets of discontinuity points of measure zero In this section we will assume that functions are defined in R with values in R. Quasicontinuous functions need not be measurable [49]. The set Q( f ) need not be measurable, however, if f : R → R is measurable, then the set Q( f ) is measurable [35]. The set Q( f ) \ C( f ) is of the first category [53], however it need not be measurable nor of measure zero. Even there is a Darboux function such that the measure of Q( f ) \C( f ) is positive [43]. If f is a quasicontinuous function then the set D( f ) is measurable as an Fσ -set, however it need not be of measure zero. Of course, if D( f ) is of measure zero then the function f is measurable. In this section we will deal with quasicontinuous functions with sets of discontinuity points of measure zero. Let `e (`) denote the outer Lebesgue measure (Lebesgue measure) in R. Denote by `e (A ∩ (x − h, x + h)) du (A, x) = lim sup 2h h→0+ the upper outer density of A ⊂ R at a point x ∈ R. Similarly, dl (A, x) = lim inf + h→0

`e (A ∩ (x − h, x + h)) 2h

is the lower outer density of A ⊂ R at a point x ∈ R. A point x ∈ R is called a density point of A ⊂ R if there exists a measurable (in the sense of Lebesgue) set B ⊂ A such that dl (B, x) = 1. The family Td of all measurable sets A such that every point x ∈ A is a density point of A is a topology called the density topology. Denote by Te the Euclidean topology on

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Ján Borsík

R. A function f is approximately continuous (at x) if it is continuous (at x) as a function f : (R, Td ) → (R, Te ). Denote by A the family of all approximately continuous functions. Approximately continuous functions need not be quasicontinuous, and quasicontinuous functions need not be approximately continuous. In [55] O’Malley introduced the topology Tae as the set of all A ∈ Td for which `(A \ Int A) = 0 and proved that f : R → R is Tae -continuous (i.e. continuous as a mapping from (R, Tae ) to (R, Te )) if and only if it is everywhere approximately continuous and almost everywhere continuous. It is easy to see that every Tae -continuous function is quasicontinuous. Denote by Cae the family of all Tae -continuous functions. Z. Grande gave the following definitions Definition 2.1. [26] A function f : R → R is s.q.c. at x if for every ε > 0 and for every U ∈ Td there is a nonempty open set V such that V ∩ U 6= 0/ and | f (y) − f (x)| < ε for all y ∈ V ∩U. Definition 2.2. [26] A function f : R → R has property A(x) at x ∈ R if there exists an open set U such that du (U, x) > 0 and the restricted function f (U ∪ {x}) is continuous at x. We will write f ∈ A(x) if f has the property A(x) at a point x. Definition 2.3. [26] A function f : R → R has property B(x) at x ∈ R (abbreviated f ∈ B(x)) if for ε > 0 we have du (Int{y : | f (y) − f (x)| < ε}, x) > 0. Denote by Qs ( f ) the set of all x at which f is s.q.c., by A( f ) the set {x ∈ R : f ∈ A(x)} and by B( f ) the set {x ∈ R : f ∈ B(x)}. Obviously, C( f ) ⊂ A( f ) ⊂ B( f ) ⊂ Qs ( f ) ⊂ Q( f ). All inclusions can be proper. However, if Qs ( f ) = R then B( f ) = R. The following theorem shows that s.q.c. functions are quasicontinuous functions with the set of discontinuity points of measure zero. Theorem 2.2. [26] The set Qs ( f )\C( f ) need not have measure zero. However, if Qs ( f ) = R then R \C( f ) = D( f ) is of measure zero. Theorem 2.3. [27] The set B( f ) \C( f ) is of measure zero. Moreover, the sets A( f ) and B( f ) have Baire property, however, they need not be borelian. Further, he gave a characterization of the set A( f ).

2. Quasicontinuous functions with small set of discontinuity points

15

Theorem 2.4. [27] Let A ⊂ R. Then A = A( f ) for some f : R → R if and only S T if A = m n Am,n , where Am,n be such that there are open sets Gn such that for each m, n ∈ N we have du (Int Am,n , x) ≥ 1/m for each x ∈ A, Am,n+1 ⊂ Am,n , Am,n ⊂ Am+1,n , Gn+1 ⊂ Gn , Gn ⊂ Am,n and du (Gn , x) ≥ 1/m for all x ∈ Am,n . Also, there exist the characterizations of the pairs (C( f ), A( f )) and (C( f ), B( f )). Theorem 2.5. [3] Let A and C be subsets of R. Then C = C( f ) and A = A( f ) for some function f : R → R if and only if there exist open sets Gn such that T C = n Gn ⊂ A, Gn+1 ⊂ Gn and inf{du (Gn , x) : n ∈ N} > 0 for each x ∈ A. Theorem 2.6. [3] Let B and C be subsets of R. Then C = C( f ) and B = B( f ) for some function f : R → R if and only if there exist open sets Gn such that T C = n Gn ⊂ B, Gn+1 ⊂ Gn and du (Gn , x) > 0 for each x ∈ B. Definition 2.4. [9] Let f : R → R be a function and let r ∈ [0, 1). We put Ar ( f ) = {x ∈ R : there is an open set U such that du (U, x) > r and f  (U ∪{x}) is continuous at x}, Alr ( f ) = {x ∈ R : there is an open set U such that dl (U, x) > r and f  (U ∪{x}) is continuous at x}, Br ( f ) = {x ∈ R : for each ε > 0 there is an open set U such that du (U, x) > r and f (U) ⊂ ( f (x) − ε, f (x) + ε)}, Blr ( f ) = {x ∈ R : for each ε > 0 there is an open set U such that dl (U, x) > r and f (U) ⊂ ( f (x) − ε, f (x) + ε)}. The set A0 ( f ) is the set A( f ) from Definition 2.2 and B0 ( f ) is B( f ) from Definition 2.3. We have Theorem 2.7. [9] Let f : R → R be a function and let 0 ≤ s < r < 1. Then

C( f )

Ar ( f ) O

/ Br ( f ) O

/ As ( f ) O

/ Al ( f ) r

/ Bl ( f ) r

/ Al ( f ) s

/ Q( f )

and each of inclusions can be proper (here, arrows mean inclusions). For r ∈ [0, 1) let Ar = { f : R → R : Ar ( f ) = R}, Arl = { f : R → R : Alr ( f ) = R}, Br = { f : R → R : Br ( f ) = R} and Brl = { f : R → R : Blr ( f ) = R}.

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Ján Borsík

Theorem 2.8. [9] Let 0 ≤ s < r < 1. Then the following inclusions hold

C

AO r

/ Br O

/ As O

/Al r

/ Bl r

/Al

/Q

s

and all inclusions are proper. According to Theorem 2.3 we have Theorem 2.9. All sets Ar ( f ) \C( f ), Arl ( f ) \C( f ), Br ( f ) \C( f ) and Brl ( f ) \ C( f ) have measure zero and all families Ar , Arl , Br and Brl have the set of discontinuity of measure zero. Moreover, for s ∈ [0, 1), the set S 1>r>s

Brl is nowhere dense set in

S

Br is nowhere dense set in As and

1>r>s Asl . So,

(Br )r∈[0,1) is the family of func-

tions between continuous functions and quasicontinuous almost everywhere continuous functions such that Br is nowhere dense subset of Bs whenever 0 ≤ s < r < 1 (in the topology of uniform convergence). Sometimes, the density of a set at a point is defined in other way.

h→0+ , k→0+

`e (A ∩ (x − h, x + h)) k+h

lim inf +

`e (A ∩ (x − h, x + h)) k+h

Du (A, x) = lim sup

Dl (A, x) =

h→0

, k→0+

Evidently, du (A, x) ≤ Du (A, x) and dl (A, x) ≥ Dl (A, x). Moreover, Dl (A, x) = 1 if and only if dl (A, x) = 1 and du (A, x) > 0 if and only if Du (A, x) > 0. More we can find in [42]. If we use in Definition 2.4 Du (U, x) and Dl (U, x) instead of du (U, x) and dl (U, x), respectively, (i.e. let D Ar ( f ) = {x ∈ R : there is an open set U such that Du (U, x) > r and f  (U ∪ {x}) is continuous at x}), and similarly D Alr ( f ), l l l D Br ( f ), D Br ( f ), D Ar , D Ar , D Br and D Ar , the corresponding Theorems 2.7, 2.8 as well as all remarks remain true, although the classes of functions are different (we have Ar ( f ) ⊂

D Ar ( f )

and Ar ⊂

D Ar ,

with equality only for r = 0). We can use in Definition 2.4 measurable sets instead of open sets.

2. Quasicontinuous functions with small set of discontinuity points

17

Definition 2.5. [37] Let ρ ∈ (0, 1). A function f : R → R is called ρ-upper continuous at x provided there is a measurable set E such that x ∈ E, Du (E, x) > ρ and f  E is continuous at x. If f is ρ-upper continuous at every point we say that f is ρ-upper continuous. Denote the class of all ρ-upper continuous functions by U Cρ . ρ-upper continuous functions are investigated in [37], [54], [41], [40], [42], [38], [36], [39]. Although the definition seems to be similar to Definition 2.4 and D Aρ ⊂ U Cρ , the differences are important. Functions from classes Ar , Arl , Br , Brl , D Ar , D Arl , D Br and D Brl are quasicontinuous the set of discontinuity points is of measure zero and they do not contain approximately continuous functions. Functions from classes U Cρ need not be quasicontinuous the measure of the set of discontinuity points can be positive and they contains approximately continuous functions. All classes of functions are measurable. Z. Grande in [29] has given the following definitions. Definition 2.6. [29] A function f : R → R has property s0 at a point x if for each positive ε and for each U ∈ Td containing x there is a point t ∈ C( f ) ∩U such that | f (t) − f (x)| < ε. A function f : R → R has property s1 at a point x if for each positive ε and for each U ∈ Td containing x there is an open interval I such that 0/ 6= I ∩U ⊂ C( f ) and | f (t) − f (x)| < ε for all points t ∈ I ∩U. A function f has property s0 (s1 ) if it has it at each point. Each function f having property s1 has also property s0 . Functions with properties s0 or s1 are quasicontinuous. Each function with property s0 at x is s.q.c. at this point. Moreover, a function f has property s0 if and only if it is s.q.c. Functions with property s0 have the set D( f ) of measure zero and functions with property s1 have the set D( f ) even of measure zero and nowhere dense. The characterization of sets of discontinuity points of these functions is following. Theorem 2.10. [20] A set A is the set of points of discontinuity of some function f : R → R with property s0 if and only if A is an Fσ -set of measure zero. Theorem 2.11. [20] A set A is the set of points of discontinuity of some function f : R → R with property s1 if and only if A is an Fσ -set of measure zero and for each nonempty set U ∈ Td contained in the closure of the set A, the set U ∩ A is nowhere dense in U.

18

Ján Borsík

E. Stro´nska investigated maximal families for classes of s.q.c. functions and functions with property s1 . Let X be a topological space and let F be a nonempty family of real functions defined on X. For F , we define the maximal additive class Madd (F ) as Madd (F ) = { f : X → R : f + g ∈ F for every g ∈ F }, the maximal multiplicative class Mmult (F ) as Mmult (F ) = { f : X → R : f · g ∈ F for every g ∈ F }, the maximal class with respect to maximum Mmax (F ) as Mmax (F ) = { f : X → R : max( f , g) ∈ F for every g ∈ F }, the maximal class with respect to minimum Mmin (F ) as Mmin (F ) = { f : X → R : min( f , g) ∈ F for every g ∈ F }, and the maximal latticelike class Mlatt (F ) as Mlatt (F ) = { f : X → R : max( f , g) ∈ F and min( f , g) ∈ F for every g ∈ F }. She proved (Qs is the family of all s.q.c. functions and Qs1 is the family of all functions with property s1 ) Theorem 2.12. [60] Madd (Qs ) = Mmax (Qs ) = Mmin (Qs ) = Mlatt (Qs ) = Qs ∩ Cae and Madd (Qs1 ) = Mmax (Qs1 ) = Mmin (Qs1 ) = Mlatt (Qs1 ) = Qs1 ∩ Cae . Let MQ denote the family of all functions with this property: if f is not Tae -continuous at x ∈ R then f (x) = 0 and du ({t ∈ R; f (t) = 0}, x) > 0. Theorem 2.13. [60] Mmult (Qs ) = Qs ∩ MQ , Mmult (Qs1 ) = Qs1 ∩ MQ and Mmult (Qs2 ) = Qs2 ∩ MQ .

2.4 Quasicontinuous functions with sets of discontinuity points almost of measure zero Z. Grande in [23] gave the following definition (` is the Lebesgue measure in Rn ). Definition 2.7. [23] A function f : Rn → R is R-integrally quasicontinuous at a point x if for each positive ε and for each open set U containing x there is a bounded Jordan measurable set I with nonempty interior such that I ⊂ U, the restricted function f  I is integrable in the sense of Riemann and R I f (t)dt `(I) − f (x) < ε. A function f is R-integrally quasicontinuous if it is such at each point.

2. Quasicontinuous functions with small set of discontinuity points

19

Theorem 2.14. [23] If a function f : Rn → R is R-integrally quasicontinuous then there is a dense open set U ⊂ Rn such that `(U \C( f )) = 0. Therefore the measure of D( f ) is zero on some dense open set. However, there is a R-integrally quasicontinuous nonmeasurable function f : R → R. Evidently, for such a function the measure of D( f ) is positive. Obviously, R-integrally quasicontinuous functions are between continuous and quasicontinuous functions. There are quasicontinuous functions which are not Rintegrally-quasicontinuous. Theorem 2.15. [23] If f : Rn → R is quasicontinuous and if there is a dense open set G ⊂ Rn such that `(G \C( f )) = 0 then f is R-integrally quasicontinuous. Therefore, in the family of almost everywhere continuous functions, quasicontinuous and R-integrally quasicontinuous functions coincide.

2.5 Quasicontinuous functions with σ -porous set of discontinuity points The notion of a σ -porous set was introduced in [15]. For a set A ⊂ R and an open interval I ⊂ R let Λ (A, I) denote the length of the largest subinterval of I having an empty intersection with A. Let x ∈ R. Then the right-porosity of the set A at x is defined as p+ (A, x0 ) = lim sup h→0+

Λ (A, (x, x + h)) , h

the left-porosity of the set A at x is defined as p− (A, x0 ) = lim sup h→0+

Λ (A, (x − h, x)) , h

and the porosity of the set A at x is defined as  p(A, x0 ) = max p− (A, x0 ), p+ (A, x0 ) . The set A ⊂ R is called right-porous at a point x ∈ R if p+ (A, x) > 0, left-porous at a point x ∈ R if p− (A, x) > 0 and porous at a point x ∈ R if p(A, x) > 0. The set A ⊂ R is called porous if A is porous at each point x ∈ A and A ⊂ R is called σ -porous if A is the countable union of porous sets.

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Every σ -porous set is of first category and of measure zero, but there are sets of first category and of measure zero, which are not σ -porous [64]. Definition 2.8. A point x ∈ R is called a point of πr -density of a set A ⊂ R for 0 ≤ r < 1 (µr -density of a set A ⊂ R for 0 < r ≤ 1) if p (R \ A, x) > r, (p (R \ A, x) ≥ r). Definition 2.9. [12] Let r ∈ [0, 1). The function f : R → R is called Pr -continuous at a point x if there exists a set A ⊂ R such that x ∈ A, x is a point of πr -density of A and f  A is continuous at a point x, Sr -continuous at a point x if for each ε > 0 there exists a set A ⊂ R such that x ∈ A, x is a point of πr -density of A and f (A) ⊂ ( f (x) − ε, f (x) + ε). Let r ∈ (0, 1]. The function f : R → R is called Mr -continuous at a point x, if there exists a set A ⊂ R such that x ∈ A, x is a point of µr -density of A and f  A is continuous at a point x, Nr -continuous at a point x, if for each ε > 0 there exists a set A ⊂ R such that x ∈ A, x is point of µr -density of A and f (A) ⊂ ( f (x) − ε, f (x) + ε). All of these functions are called porouscontinuous functions. Symbols Pr ( f ), Sr ( f ), Mr ( f ) and Nr ( f ) will denote the sets of all points at which the function f is Pr -continuous, Sr -continuous, Mr -continuous and Nr -continuous, respectively. Collectively, these sets will be called the sets of porouscontinuity points of the function f . Porouscontinuity was defined by the set A containing the point x. There is, however, a second option using an open set B where the continuity would be required at a point x for f  B ∪ {x}. In [12] it is shown that it results in the same notion. This is a difference with the measure case. Theorem 2.16. [12] Let f : R → R. Then the set S0 ( f ) \ C ( f ) is σ -porous. The following theorem summarizes relations between sets of continuity, porouscontinuity and quasicontinuity of a function f : R → R. Theorem 2.17. [12] Let 0 < r < s < 1 and f : R → R. Then C ( f ) ⊂ M1 ( f ) = N1 ( f ) ⊂ Ps ( f ) ⊂ Ss ( f ) ⊂ Ms ( f ) = Ns ( f ) ⊂ Pr ( f ) ⊂ P0 ( f ) ⊂ S0 ( f ) ⊂ Q( f ). All inclusions are proper. Let there be introduced the following denotations: Mr = { f : Mr ( f ) = R}, Nr = { f : Nr ( f ) = R}, Pr = { f : Pr ( f ) = R}, Sr = { f : Sr ( f ) = R}.

2. Quasicontinuous functions with small set of discontinuity points

21

Theorem 2.18. Let 0 < r < s < 1. Then C ⊂ M1 = N1 ⊂ Ps ⊂ Ss ⊂ Ms = Ns ⊂ Pr ⊂ P0 ⊂ S0 ⊂ Q. All inclusions are proper. Therefore functions in the family S0 , and so all porouscontinuous functions, have σ -porous sets of discontinuity points.

2.6 Limits It is easy to see that the family of quasicontinuous functions is closed under uniform convergence. Theorem 2.19. [9] Let s ∈ [0, 1). Then the sets Br , Brl ,

S 1>r>s

Br and

S 1>r>s

Brl

are closed in the topology of the uniform convergence. However, the sets Ar and Arl are not closed. Theorem 2.20. [9] For each r ∈ [0, 1) there is a sequence ( fn )n of functions belonging to Arl such that its uniform limit does not belong to Ar . Problem 2.1. Characterize uniform limits of Ar and Arl . Is it true that each function from Br (Brl ) can be written as the uniform limit of functions from Ar (Arl )? (Z. Grande in [26] has shown that this is true for B0 .) Similarly, by [13], the families Sr and Mr are closed under uniform convergence, whereas families Pr not. The family of R-integrally quasicontinuous functions is not closed under uniform convergence [23]. Let X be a topological space and (Y, d) a metric one. We say that a sequence of functions fn : X → Y discretely converges to the function f : X → Y ([14]) if ∀x ∈ X∃n(x)∀n ≥ n(x) : fn (x) = f (x). Z. Grande in [22] has characterized discrete limits of quasicontinuous almost everywhere continuous functions. Theorem 2.21. [22] A function f : R → R is the discrete limit of a sequence of quasicontinuous almost everywhere continuous functions if and only if the set R \ Q( f ) is nowhere dense and there is an Fσ -set A of measure zero such that the restriction f  (R \ A) is the discrete limit of a sequence of continuous functions (on R \ A).

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Recall that a sequence of functions fn : X → Y quasiuniformly converges to f : X → Y if the sequence ( fn )n pointwise converges to f and ∀ε > 0∀m ∈ N∃p ∈ N∀x ∈ X : min{d( fm+1 (x), f (x)),. . ., d( fm+p (x), f (x))} < ε. The quasiuniform limit of continuous functions is continuous but the quasiuniform limit of quasicontinuous functions need not be quasicontinuous. However, the quasiuniform limit of quasicontinuous functions is cliquish. In [8] it is shown that every cliquish function f : R → R can be expressed as the quasiuniform limit of a sequence of quasicontinuous functions. The result was strengthened, by showing it holds for functions defined on more general spaces. Ch. Richter has shown [56] that this is true for functions defined on pseudometrizable spaces and by Z. Grande [21], we can assume moreover that functions are quasicontinuous and Darboux. The uniform limit of s.q.c. functions fn : R → R is s.q.c. Since s.q.c. functions have the sets of discontinuity points of measure zero (Theorem 2.2), the quasiuniform limit of sequence of s.q.c. functions has the set of discontinuity points of measure zero. Theorem 2.22. [29] A function f : R → R is almost everywhere continuous if and only if there is a sequence of Darboux s.q.c. functions quasiuniformly convergent to f . Similar result we can find for functions with property s1 . Theorem 2.23. [58] A function f : R → R is almost everywhere continuous if and only if there are functions fn : R → R with property s1 quasiuniformly converging to f Since the set of discontinuity of porouscontinuous functions is σ -porous, the quasiuniform limit of a sequence of some porouscontinuous functions has the set of discontinuity points σ -porous and previous theorem is not true for porouscontinuous functions. Problem 2.2. Is every function f : R → R with σ -porous set of points of discontinuity the quasiunform limit of a sequence (some) porouscontinuous functions?

2.7 Quasicontinuous almost everywhere continuous functions Evidently, the biggest class of quasicontinuous functions with the set of discontinuity points of measure zero is the family of almost everywhere continuous quasicontinuous functions.

2. Quasicontinuous functions with small set of discontinuity points

23

It is easy to see that the uniform limit of quasicontinuous almost everywhere continuous functions is quasicontinuous almost everywhere continuous. From Theorem 2.23 we obtain that each almost everywhere continuous function is the quasiuniform limit of a sequence of quasicontinuous almost everywhere continuous functions. Almost everywhere continuous function f : R → R has dense set of continuity, so it is cliquish. According to [10] (also [24], [47]), each cliquish function f : R → R is the sum of two quasicontinuous functions f1 and f2 such that D( f1 ) ∩ D( f2 ) ⊂ D( f ). So, immediately we have the characterization of the sums of quasicontinuous almost everywhere continuous functions. Theorem 2.24. A function f : R → R is almost everywhere continuous if and only if it is the sum of two quasicontinuous functions both with the set of discontinuity points of measure zero. However, it need not be the sum of two functions from the family Ar . Similarly, each function with σ -porous set of discontinuity is the sum of two quasicontinuous functions with σ -porous set of discontinuity points.

2.8 Other classes of functions between continuous and quasicontinuous functions Of course, each family of quasicontinuous functions with some extra property lies between continuous and quasicontinuous functions. For example, Darboux ´ ¸ tkowski and quasicontinuous functions (see survey paper [51]), strong Swia ´ functions (e.g. [47], [61]), extra strong Swia¸tkowski functions [62], which are both Darboux and quasicontinuous however their set of dicontinuity points can be of positive measure. Quasicontinuous functions with closed graph [11], [57] or internally quasicontinuous functions (a function f is internally quasicontinuous [48] if is quasicontinuous and its set of points of discontinuity is nowhere dense) are such that the set of discontinuity is nowhere dense, but it can be of positive measure. However, it is a subject for another paper. Acknowledgements. The paper was supported by Grant VEGA 2/0050/15 and APVV-0269-11.

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References [1] R. Baire, Sur les functions des variables reelles, Ann. Mat. Pura Appl. 3 (1899), 1-122. [2] W. W. Bledsoe, Neighborly functions, Proc. Amer. Math. Soc. 3 (1952), 114-115. [3] J. Borsík, On strong quasicontinuity and continuity points, Tatra Mt. Math. Publ. 30 (2005), 47-57. [4] J. Borsík, Points of continuity and quasicontinuity, Cent. Eur. J. Math. 8 (2010), 179190. [5] J. Borsík, Points of continuity, quasicontinuity and cliquishness, Rend. Ist. Matem. Univ. Trieste 26 (1994), 5-20. [6] J. Borsík, Points of continuity, quasicontinuity, cliquishness and upper and lower quasicontinuity, Real Anal. Exchange 33 (2007/08), 339-350. [7] J. Borsík, Points of generalized continuities, Tatra Mt. Math. Publ. 52 (2012), 151158. [8] J. Borsík, Quasiuniform limits of quasicontinuous functions, Math. Slovaca 42 (1992), 269-274. [9] J. Borsík, Some classes of strongly quasicontinuous functions, Real Anal. Exchange 30 (2004/05), 689-702. [10] J. Borsík, Sums of quasicontinuous functions defined on pseudometrizable spaces. Real Anal. Exchange 22 (1996/97), 328-337. [11] J. Borsík, J. Doboš, M. Repický, Sums of quasicontinuous functions with closed graphs, Real Anal. Exchange 25 (1999/2000), 679-690. [12] J. Borsík, J. Holos, Some properties of porouscontinuous functions, Math. Slovaca 64 (2014), 741-750. [13] J. Borsík, J. Holos, Some remarks on porouscontinuous functions, in preparation [14] Á. Császár, M. Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar. 10 (1975), 463-472. [15] E. P. Dolženko, Boundary properties of arbitrary functions, (in Russian), Math. USSR Izv. 31 (1967), 3-14. [16] J. Ewert, J. S. Lipi´nski, On points of continuity, quasi-continuity and cliquishness of maps, Topology Appl., 5th Colloq. Eger 1983, Colloq. Math. Soc. János Bolyai 41 (1985), 269-281. [17] J. Ewert, J. S. Lipi´nski, On points of continuity, quasicontinuity and cliquishness of real functions, Real Anal. Exchange 8 (1982/83), 473-478. [18] J. Ewert, J. S. Lipi´nski, On relations between continuity, quasi-continuity and cliquishness of maps, General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp. Prague 1986, Res. Expo. Math. 16 (1988), 177-185. [19] E. Grande, Sur un probléme concernat les fonctions quasicontinues, Math. Slovaca 32 (1982), 309-312. [20] M. Grande, On the sets of discontinuity points of functions satisfying some approximate quasi-continuity conditions, Real Anal. Exchange 27 (2001/02), 773-782. [21] Z. Grande, On Borsík’s problem concerning quasiuniform limits of Darboux quasicontinuous functions, Math. Slovaca 44 (1994), 297-301. [22] Z. Grande, On discrete limits of sequences of approximately continuous functions and Tae -continuous functions, Acta Math. Hungar. 92 (2001), 39-50. [23] Z. Grande, On Riemann integral quasicontinuity, Real Anal. Exchange 31 (2005/06), 239-252.

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[24] Z. Grande, On some represetations of a.e. continuous functions, Real Anal. Exchange 21 b(1995/96), 175-180. [25] Z. Grande, On some special notions of approximative quasi-continuity, Real Anal. Exchange 24 (1998/99), 171-183. [26] Z. Grande, On strong quasi-continuity of functions of two variables, Real Anal. Exchange 21 (1995/96), 236-243. [27] Z. Grande, On strong quasi-continuity points, Tatra Mt. Math. Publ. 8 (1996), 17-21. [28] Z. Grande, Quasicontinuity, cliquishness and the Baire property of functions of two variables, Tatra Mt. Math. Publ. 24 (2002), 29-35. [29] Z. Grande, On some special notions of approximate quasi-continuity, Real Anal. Exchange 24 (1998/99), 171-184. [30] Z. Grande, On quasi-uniform convergence of sequence of s.q.c. functions, Math. Slovaca 48 (1998), 507-511. [31] Z. Grande, On the maximal families for the class of strongly quasicontinuous functions, Real Anal. Exchange 20 (1994/95), 631-638. [32] Z. Grande, On the maximal multiplicative family for the class of quasicontinuous functions, Real Anal. Exchange 15 (1989/90), 437-441. [33] Z. Grande, Sur la quasi-continuité et la quasi-continuité approximative, Fund. Math. 129 (1988), 167-172. [34] S. Kempisty, Sur les fonctions quasicontinued, Fund. Math. 19 (1932), 184-197. [35] P. Kostyrko, Quasicontinuity and some classes of Baire 1 functions, Commentat. Mat. Univ. Carol. 29 (1988), 601-609. [36] S. Kowalczyk, Compositions of ρ-upper continuous functions, Math. Slovaca (to appear) [37] S. Kowalczyk, K. Nowakowska, A note on the ρ-upper continuous functions, Tatra Mt. Math. Publ. 44 (2009), 153-158. [38] S. Kowalczyk, K. Nowakowska, A note on the [0]-lower continuous functions, Tatra Mt. Math. Publ. 58 (2014), 111-128. [39] S. Kowalczyk, K. Nowakowska, On O’Malley ρ-upper-continuous functions, Math. Slovaca (submitted) [40] S. Kowalczyk, K. Nowakowska, Maximal classes for ρ-upper continuous functions, J. Appl. Anal. 19 (2013), 69-89. [41] S. Kowalczyk, K. Nowakowska, Maximal classes for the family of [λ , ρ]-continuous functions, Real Anal. Exchange 36 (2010/2011), 307-324. [42] S. Kowalczyk, K. Nowakowska, Path continuity connected with the notion of density, Traditional and present-day topics in real analysis, University of Łód´z, Łód´z 2013, 449-471. ´ atkowski [43] J. Kucner, R. J. Pawlak, On local characterization of the strong Swi ˛ property for a function f : [a, b] → R, Real Anal. Exchange 28 (2002/03), 563-572. [44] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41. [45] J. S. Lipi´nski, T. Šalát, On the points of quasi-continuity and cliquishness of functions, Czechoslovak Math. J. 21 (1971), 484-489. [46] A. Maliszewski, Darboux property and quasi-continuity. A uniform approach, WSP, Słupsk, 1996. [47] A. Maliszewski, Sums and products of quasi-continuous functions, Real Anal. Exchange 21 (1995/96), 320-329.

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[48] M. Marciniak, P. Szczuka, Maximums of internally quasi-continnuous functions, Tatra Mt. Math. Publ. 52 (2012), 83-90. [49] S. Marcus, Sur les fonctions quasicontinues au sens de S. Kempisty, Colloq. Math. 8 (1961), 47-53. [50] O. V. Maslyuchenko, The discontinuity points sets of quasi-continuous functions, Bull. Austral. Math. Soc. 75 (2005), 373-379. [51] T. Natkaniec, On quasi-continuous functions having Darboux property, Math. Pannon. 3 (1992), 81-96. [52] T. Neubrunn, Quasicontinuity, Real Anal. Exchange 14 (1988/89), 259-306. ˇ [53] A. Neubrunnová, On quasicontinuous and cliquish functions, Casopis Pˇest. Mat. 99 (1974), 109-114. [54] K. Nowakowska, On a family of [λ , ρ]-continuous functions, Tatra Mt. Math. Publ. 44 (2009), 129-138. [55] R. J. O’Malley, Approximately differentiable functions. The r topology, Pacific J. Math. 72 (1977), 207-222. [56] Ch. Richter, Representing cliquish functions as quasiuniform limits of quasicontinuous functions, Real Anal. Exchange 27 (2001/2002), 209-221. [57] W. Sieg, Maximal classes for the family of quasi-continuous functions with closed graph, Demonstratio Math. 42 (2009), 41-45. [58] E. Stro´nska, On quasi-uniform convergence of sequences of s1 -strongly quasicontinuous functions on Rm , Real Anal. Exchange 30 (2004/2005), 217-234. [59] E. Stro´nska, On the maximal families for some classes of strongly quasicontinuous functions on Rm , Real Anal. Exchange 32 (2006/2007), 3-18. [60] E. Stro´nska, On the maximal families for some special classes of strongly quasicontinuous functions, Real Anal. Exchange 23 (1997/98), 743-752. ´ atkowski [61] P. Szczuka, Maximal classes for the family of strong Swi ˛ functions, Real Anal. Exchange 28 (2002/2003), 429-437. ´ atkowski [62] P. Szczuka, Sums and products of extra strong Swi ˛ functions, Tatra Mt. Math. Publ. 49 (2011), 71-79. [63] H. P. Thielman, Types of functions, Amer. Math. Monthly 60 (1953), 158-161. [64] L. Zajíˇcek, Porosity and σ -porosity, Real Anal. Exchange 13 (1987/88), 314-350.

JÁN B ORSÍK Mathematical Institute, Slovak Academy of Sciences Grešákova 6, 04001 Košice, Slovakia Katedra fyziky, matematiky a techniky FHPV, Prešovská univerzita v Prešove ul. 17. novembra 1, 08001 Prešov, Slovakia E-mail: [email protected]

Chapter 3

Convolution operators on some spaces of functions and distributions in the theory of circuits

´ ANDRZEJ BORYS, ANDRZEJ KAMINSKI AND SŁAWOMIR SOREK

2010 Mathematics Subject Classification: 94C05, 47B38, 46F10. Key words and phrases: linear circuit, nonlinear circuit, Taylor series, Volterra series, (neutrix) product of distributions, (neutrix) kth power of a distribution, convolution of kth order of functions, (neutrix) convolution of kth order of distributions.

3.1 Introduction For the theory of linear systems and circuits, investigated in telecommunications, electronics and signal processing [10, 12], the Dirac delta impulse δ is a natural and useful object but no function in the classical sense corresponds to it. The notion can be mathematically justified on the base of the theory of distributions created by L. Schwartz [21] and appears to be very fruitful in various fields of applications and in mathematics itself. In particular, the signal δ = δ (t), meant as a distribution (generalized function) of time t on the real line R, allows one to determine in some cases the input-output characteristics of a non-autonomous linear system as well as its impulse response in the theory of systems and circuits. We present here our attempt to deliver a strict mathematical basis for some aspects of the theory of linear and nonlinear systems and circuits extending the domain of objects in use from functions to distributions to embrace δ , in particular. The presented work was inspired by the talk [5] delivered by the first

28

Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

author during the conference on generalized functions in Be¸dlewo in 2007. We recall the results of the present authors given in [4] and in [16], extending them in section 7 by one of the results of the third author which are going to be published separately (see [4]). The basic notation and the definition of the convolution of kth order, the notion crucial for our considerations, are given in section 2. In section 3, the two types of linear circuits are discussed: circuits with decaying memory and memoryless ones. We describe them in terms of linear operators defined for functions and extended suitably for distributions. This extension allows one to represent both types of circuits as convolution operators determined by the impulse response distributions. In section 4, we extend the theory to the nonlinear Volterra systems described by Volterra and Taylor series and discuss conditions under which the corresponding nonlinear operators are well defined on certain spaces of functions. A possibility of defining these nonlinear operators for the Dirac delta impulse, desirable for applications but not attainable in the standard sense of operations on distributions, was posed as a problem in [5]. Two aspects of the problem, concerning the product of k distributions (in particular, the kth power of the Dirac delta) and the convolution of kth order of distributions (in particular, the kth convolution of the Dirac delta) are discussed and solved in sections 6 and 7 by means of the notion of neutrix. A general concept of neitrix was introduced by J. G. van der Corput in [7] and then it was adapted in a particular form to the product and the convolution of distributions by B. Fisher and his co-authors in numerous papers, but their approach contains certain mathematical incoherences (see Remarks 3.4 and 3.5). Therefore we discuss in section 5 some aspects of the theory more carefully and remove its drawbacks due to certain essential modifications and generalizations. In particular, we replace Fisher’s neutrix of sequences by the corresponding neutrix of nets. In our opinion, this is a good example of the situation where nets appear to be a more adequate tool in the theory of the product of generalized functions (see also [6, 20]). In section 6, we present a solution to the first part of the problem, concerning the product of distributions. Following the ideas of E. L. Koh and C. K. Li in [17], we show how to define the kth power of the Dirac delta distribution and, more generally, the product of k distributions, in the sense of the notion of neutrix suitably modified in section 5. We prove the result of Koh and Li for a certain net neutrix.

3. Convolution operators in the theory of circuits

29

Theorem 3.2 given in section 7 is an answer to the second part of the problem, concerning the kth convolution of distributions, also in the sense of net neutrix discussed in section 5. A complete proof of Theorem 3.2 and other aspects of the theory are discussed in [22].

3.2 Basic definitions and notation The symbols N, N0 and R denote the sets of all positive integers, all nonnegative integers and all real numbers, respectively. For given j ∈ N the symbols N0j and R j denote the Cartesian products of j copies of the sets N0 and R, respectively; in particular, the symbol R jk for j, k ∈ N means the Cartesian product of k copies of R j . The expressions: measurable functions, almost everywhere, almost all are meant in the sense of Lebesgue. We will start with considering certain convolution operators on the spaces L1 (R j ) and L1 (R jk ) of integrable functions on R j and R jk as well as on the spaces L∞ (R j ) and L∞ (R jk ) of essentially bounded functions on R j and R jk , respectively, but later we will extend our considerations for spaces D 0 of distributions and S 0 of tempered distributions defined on the Euclidean space of a suitable dimension. Let us recall that the space D 0 = D 0 (R j ) of distributions on R j is the strong dual of the space D = D(R j ) of test functions on R j , i.e. smooth (infinitely differentiable) functions of compact support, endowed with the respective inductive limit topology, while the space S 0 = S 0 (R j ) of tempered distributions, a subspace of D 0 = D 0 (R j ), is the dual of the space S = S (R j ) of smooth functions rapidly decreasing together with all derivatives at infinity, endowed with the respective metric topology (see [21]; see also [1] and [13]). The space D 0 of distributions contains regular distributions corresponding to locally integrable functions and, in particular, to members of the spaces L1 and L∞ . Important examples of (tempered) distributions on R j (which are not represented by usual functions) are the Dirac delta, that we denote by δ or by δ( j) to mark the dimension of R j , defined as follows: < δ , ϕ >=< δ( j) , ϕ >:= ϕ(0),

ϕ ∈ D(R j ) (ϕ ∈ S (R j )) (l)

as well as its distributional derivatives δ (l) = δ( j) defined by < δ (l) , ϕ >:= (−1)l ϕ (l) (0),

ϕ ∈ D(R j ) (ϕ ∈ S (R j ))

(3.1)

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Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

for arbitrary l ∈ N0j , according to the standard multidimensional notation. The following modification of the convolution of functions plays an important role in further considerations concerning nonlinear circuits. Definition 3.1. Let j, k ∈ N. Assume that f : R j → R and h : R jk → R are measurable functions. By the convolution of kth order or shortly the kth convoluk

tion of the functions h and f we mean the measurable function h ∗ f : R j → R defined almost everywhere on R j by the following k-multiple integral: k

(h ∗ f )(x) :=

Z R jk

h(η1 , . . . , ηk ) f (x − η1 ) · . . . · f (x − ηk ) dη1 . . . dηk , (3.2)

where x, η1 , . . . , ηk ∈ R j , under the condition that Z R jk

|h(η1 , . . . , ηk ) f (x − η1 ) · . . . · f (x − ηk )| dη1 . . . dηk < ∞

for almost all x ∈ R j . k

Remark 3.1. Assume that h ∈ L1 (R jk ). If f ∈ L1 (R j ), then h ∗ f ∈ L1 (R j ), due k

to the Fubini theorem. On the other hand, if f ∈ L∞ (R j ), then h ∗ f ∈ L∞ (R j ). k

Assume now that h ∈ L∞ (R jk ). If f ∈ L1 (R j ), then h ∗ f ∈ L∞ (R j ). But if k

f ∈ L∞ (R j ), then the convolution of kth order h ∗ f need not exist, e.g. in case f and h are constantly equal to 1 on R j and R jk , respectively. 1

Clearly, h ∗ f = h ∗ f , where h ∗ f means the classical convolution of the functions h and f .

3.3 Linear circuits For simplicity we will assume further on that j = 1 and k ∈ N. To describe a linear circuit one usually assumes that an input signal x = x(t) and an output signal y = y(t), functions of time t ∈ R, are related to each other by a black box linear operator L, i.e. a convolution operator of the form: y = Lx = h ∗ x for a certain function h = h(t), interpreted as a circuit impulse response. This is schematically shown on Fig. 1.

3. Convolution operators in the theory of circuits

31

black box input

output -

x = x(t)

-

linear operator L : x 7→ y; y(t) = (Lx)(t) = (h ∗ x)(t)

y = y(t)

Fig. 1. Scheme of linear circuit

In linear circuits with decaying memory, one usually assumes that h is an integrable function on R and L = Lm is the convolution operator given in the two cases: (a) for x ∈ L1 (R), (b) for x ∈ L∞ (R), by the same formula: Z+∞

y(t) = (Lm x)(t) = (h ∗ x)(t) =

h(τ)x(t − τ) dτ,

t ∈ R,

(3.3)

−∞

i.e. Lm maps the input signals: (a) x ∈ L1 (R), (b) x ∈ L∞ (R) to the output signals: (a) y ∈ L1 (R), (b) y ∈ L∞ (R), respectively. In other words, Lm is the convolution operator acting in the two considered cases as follows: (a) Lm : L1 (R) → L1 (R),

(b) Lm : L∞ (R) → L∞ (R).

It is known that L1 (R) is a convolution algebra without unit, but the Dirac delta plays the role of the convolution unit in the wider space D 0 of distributions with the convolution meant in the more general distributional sense (see [1, 21]): f ∗δ = δ ∗ f = f, f ∈ D 0. (3.4) The operator Lm , defined in (3.3) in case the input signal x and the impulse response h are functions, can be extended to include both x = δ and h = δ . If h = δ , due to (3.4), one extends Lm to the linear operator Lm : D 0 → D 0 of the form: y = Lm x = δ ∗ x = x, x ∈ D 0. (3.5) If x = δ , the extension of Lm makes sense for every h ∈ D 0 and has the form: y = Lm δ = h ∗ δ = h,

h ∈ D 0,

(3.6)

which is particularly useful, because the output signal and the impulse response are then equal, i.e. a system is fully described by its impulse response. In par-

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Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

ticular, if x = δ and h = δ , then both (3.5) and (3.6) yield Lm δ = δ ∗ δ = δ . We use the same symbol Lm for the extended linear operator given by (3.5) and (3.6) and for that originally defined in (3.3), because the extension is consistent. Another type of linear systems, without memory, considered in electrical engineering and telecommunications as linear memoryless systems (circuits), may be described by a linear operator L = Lnm of the form: α ∈ R,

y = Lnm x = αx,

(3.7)

where the input signals x are, as in (3.3), functions from a given space or, as in (3.5), distributions. The latter is more general and Lnm : D 0 → D 0 given by (3.7) can be expressed in the form: α ∈ R, x ∈ D 0 ,

y = Lnm x = αx = h0 ∗ x,

(3.8)

where h0 := αδ , because αx = α(δ ∗ x) = (αδ ) ∗ x,

x ∈ D 0,

in view of (3.4). Hence Lnm in (3.8) can be treated as an input-output description of a memoryless circuit in the form of the extended convolution operator with the impulse response h0 = αδ ∈ D 0 . An example of a memoryless circuit is a simple resistive voltage divider, consisting of two resistors R1 and R2 , presented on Fig. 2.

◦ 6

R1 •

x = x(t)

◦ 6

y = y(t)

R2 ◦





Fig. 2. Example of resistive voltage divider

3. Convolution operators in the theory of circuits

33

The linear memoryless operator Lnm has the form: y = Lnm x =

R2 x = h0 ∗ x, R1 + R2

where

R2 δ. R1 + R2 Combining a memoryless circuit and a circuit with (decaying) memory, described by Lnm and Lm , we see that the linear operator Lov := Lnm + Lm describing the overall circuit is, by (3.3), (3.5) and (3.8), of the following form: h0 :=

x ∈ D 0,

y = Lov x = Lnm x + Lm x = αx + h ∗ x = hov ∗ x,

(3.9)

where hov := h0 + h = αδ + h ∈ D 0 is the impulse response of the overall circuit.

3.4 Nonlinear circuits An important class of nonlinear circuits, studied e.g. in [2, 3, 11], is described by the Taylor power series and Volterra series, i.e. the formal series of the form: ∞ ∞ y = Tnm x =

∑ αk x k ,

y = Tm x =

k=1

k

∑ hk ∗ x,

(3.10)

k=1

where αk ∈ R, x = x(t) is a function on R, hk = hk (t1 , . . . ,tk ) are functions k

on Rk and hk ∗ x are functions on R described in Definition 3.1 such that all expressions in (3.10) and (3.2) are well defined. The nonlinear mappings Tnm and Tm , defined by (3.10), as well as Tov of the form: ∞

Tov x = (Tnm + Tm ) x =



k

∑ αk xk + ∑ hk ∗ x, k=1

(3.11)

k=1

are extensions of the linear operators Lnm , Lm and Lov , defined for suitable functions x in (3.3), (3.7) and (3.9), respectively (and coincide with them, respectively, if αk = 0 and hk = 0 for k ≥ 2). The functions hk on Rk are called the linear (for k = 1) and nonlinear (for k > 1) impulse responses of the kth order.

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Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

Obviously, to have the mappings Tnm , Tm and Tov well defined, one has to impose some assumptions on the right-hand sides of (3.10) and (3.11). Assume that hk ∈ L1 (Rk ) for k ∈ N and consider the two cases: (a) x ∈ L1 (R), (b) x ∈ L∞ (R). In both cases, each member of the second series in (3.10) is well k

k

defined and, by the Fubini theorem, hk ∗ x ∈ L1 (R) in case (a), and hk ∗ x ∈ L∞ (R) in case (b). Additional assumptions concerning convergence of the two series in (3.10) are necessary. In case (b), the mappings Tnm , Tm and Tov are well defined if both series are convergent uniformly, i.e. in L∞ . In case (a), Tm is well defined if the second series in (3.10) is convergent in L1 (R), but to have the mapping Tnm well defined assume, in addition, that x ∈ Lk (R) for all k ∈ N and the first series in (3.10) is convergent in L1 (R). We may try to extend the nonlinear mappings Tnm , Tm and Tov , as it was done for their linear counterparts in the preceding section, from the above particular spaces of functions to distributions, at least in the special case of the input signal x and the impulse response hk are the Dirac delta distributions: x = δ = δ(1) on R and hk = δ(k) on Rk , respectively (according to the notation introduced in (3.1)). However, we then encounter mathematical difficulties: putting x = δ in the first series in (3.10) for k > 1 is not allowed, because the power δ k of δ for k > 1 does not exist in the standard sense of the theory of distributions (for k

k = 2 see e.g. [1], pp. 243-244); a similar difficulty concerns δ(k) ∗ δ in the second series, where δ(k) is the Dirac delta on Rk . How to overcome these two difficulties was asked in [5]. In section 3.6, we present a solution to the first part of the problem and in section 3.7 to the k

second one: both the power δ k of δ and the convolution of kth order δ(k) ∗ δ exist in the sense of net neutrix described in the next section.

3.5 Neutrices We start from recalling van der Corput’s general definition of neutrix given in [7]. Then we impose Assumptions 3.1 and 3.2 used in the sequel and specify the form of neutrices used in the theory of the product of distributions. Definition 3.2. Let N 0 be an arbitrary nonempty set and N 00 be a commutative additive group. By a neutrix (of type (N 0 , N 00 )) one means a commutative additive group N of functions ν: N 0 → N 00 (called negligible functions) such that (∗) the only constant function ν in N is ν ≡ 0.

3. Convolution operators in the theory of circuits

35

Condition (∗) guarantees the uniqueness of N−limits in the sense of the N−convergence, defined by means of the neutrix N in the following way: Definition 3.3. Let N 0 be a nonempty subset of a certain set N 0 , let a be a fixed element of N 0 and assume that ξ → a is well defined for ξ ∈ N 0 , e.g. N 0 is a subset of a topological space N 0 and a ∈ N 0 is a limit point of N 0 . Moreover, assume that N 00 is a commutative additive group and N is a neutrix in the sense of Definition 3.2. For ν: N 0 → N 00 and l ∈ N 00 , we define N−lim ν(ξ ) = l, ξ →a

if ν0 ∈ N,

(3.12)

where ν0 (ξ ) := ν(ξ ) − l,

ξ ∈ N 0.

Clearly, if N1 and N2 are neutrices as in Definition 3.3 such that N1 ⊆ N2 , then N1−convergence implies N2−convergence. Proposition 3.1. Let N 0 := (0, 1), N 0 := [0, 1] and a := 0 ∈ N 0 be a limit point of (0, 1) in the standard topology of [0, 1]. Assume that N 00 := X is a topological vector space (over R) and fix a neutrix N := N X of type ((0, 1), X), i.e. a commutative additive group of γ ∈ X (0,1) satisfying (∗). We call γ = (γ τ ) = (γ τ )τ∈(0,1) nets in X. If N 0 and N 0 above are replaced by N 0 := N, N 0 := N ∪ {∞}, a fixed neutrix N of type (N, X) of sequences (γn ) ∈ X N satisfying (∗) will be denoted by NX . Formula (3.12) defines the neutrix limits N X−lim γ τ and NX− lim γn in X for all nets γ = (γ τ ) ∈ X (0,1) and all sequences τ→0

n→∞

(γn ) ∈ X N , respectively. Denote by c0 (X) the set of all nets α = (α τ ) ∈ X (0,1) convergent to 0 as τ → 0 and by c0 (X) the set of all sequences (αn ) ∈ X N convergent to 0 as n → ∞ in the topology of X; if X = R we write c0 := c0 (R) and c0 := c0 (R). By d ∞ denote the set of all nets β = (β τ ) ∈ R(0,1) divergent to ∞ as τ → 0 and by d∞ the set of all sequences (βn ) ∈ RN divergent to ∞ as n → ∞. Clearly, α = (α τ ) ∈ c0 (X) iff (αn ) ∈ c0 (X) for all (αn ) of the form αn := α τn , τn ∈ (0, 1), τn → 0 and β = (β τ ) ∈ d ∞ iff (βn ) ∈ d∞ for all (βn ) of the form βn := β τn , τn ∈ (0, 1), τn → 0. Remark 3.2. The convergence of nets (sequences) in the topology of X implies the N X−convergence (resp. NX−convergence) to the same limit iff N X ⊇ c0 (X) (resp. NX ⊇ c0 (X)); they coincide if the equality holds in the inclusion, so to extend essentially the respective neutrix convergence one has to add nets (sequences) not convergent to 0 in X to the neutrix. For example, if X = R,

36

Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

it is standard to assume [7, 9] that a given neutrix N R (resp. NR ) contains c0 (resp. c0 ) and a certain subclass d ∗ of d ∞ (resp. d∗ of d∞ ) which determines ∗ the neutrix N R = N d (resp. NR = Nd∗ ) in the following way: all negligible ∗ functions in N d (resp. Nd∗ ) are finite linear sums of elements of c0 and d ∗ (resp. c0 and d∗ ). That means, ∗

N d := span(c0 ∪ d ∗ );

Nd∗ := span(c0 ∪ d∗ ).

(3.13)



The range of the extensions of the N d −convergence and Nd∗−convergence depends essentially on the selection of the subclasses d ∗ and d∗ of d ∞ and d∞ , respectively. If X = E 0 is the dual of a topological vector space E (over R) endowed with the weak topology, then it is natural to define the corresponding neutrices N X and NX via given neutrices N R and NR , by means of values of x0 ∈ E 0 on x ∈ E. Proposition 3.2. Assume that E is a topological vector space (over R) and X := E 0 is its dual endowed with the weak topology. Under Assumption 3.1, define the neutrix N X generated by N R as follows: (γ τ ) ∈ N X if (hγ τ , xi) ∈ N R for x ∈ E. Obviously, for γ = (γ τ ) ∈ X (0,1) and γ ∗ ∈ X, we have N X − lim γ τ = τ→0



γ ∗ iff N R− lim hγ τ , xi = hγ ∗ , xi for x ∈ E. Denote, in particular, by N X,d the τ→0



neutrix generated by N d of the form (3.13). Similarly, we define the neutrices NX , NX,d∗ , generated by given neutrices NR , NR,d∗ , and the respective neutrix convergences. Remark 3.3. In particular, if X = D 0 (X = S 0 ) in Assumption 3.2, we have ∗ ∗ ( f τ ) ∈ N X,d iff (h f τ , ϕi) ∈ N d for ϕ ∈ E and, consequently, ∗

N X,d −lim f τ = f τ→0



iff N d −lim h f τ , ϕi = h f , ϕi for

ϕ ∈ E,

τ→0

where E = D (E = S ), respectively. Analogously, we define the neutrix convergence Nd∗− lim fn in D 0 (in S 0 ). Thus the neutrix convergences in D 0 and n→∞

in S 0 are determined by a suitable choice of the classes d ∗ and d∗ . B. Fisher in [9] has chosen and used in his numerous papers on neutrix products and convolutions of distributions the fixed neutrix of sequences defined by the class d∗ := dF , where dF consists of all sequences (βn )n∈N whose members are of the form: βn := nλ lnr n

for λ > 0, r ∈ N0

or

λ = 0, r ∈ N.

(3.14)

3. Convolution operators in the theory of circuits

37

Consider the corresponding neutrix of nets defined by d ∗ := d F , where d F consists of all nets (β τ )τ∈(0,1) whose members are of the form: β τ := τ −λ (− ln τ)r

for λ > 0, r ∈ N0

or

λ = 0, r ∈ N.

(3.15)

To prove Theorem 3.1 and 3.2 in a stronger form, consider also the following narrower class d ∗ := d P , where d P consists of all nets (β τ )τ∈(0,1) whose members are of the form: β τ := τ −r

for r ∈ N.

(3.16)

Definition 3.4. Denote by NF the sequential neutrix of Fisher defined by the equality on the right hand side of (3.13) with d∗ = dF given by (3.14). On the other hand, denote by N F and N P the net neutrices defined by the equality on the left hand side of of (3.13) with d ∗ = d F and d ∗ = d P given by (3.15) and (3.16), respectively. Clearly, the neutrix N P is essentially narrower than the neutrix N F . Remark 3.4. The sequential neutrix NF of Fisher has an essential drawback. Namely, a subsequence of a sequence belonging to NF does not belong to NF , in general. Consequently, a subsequence of a sequence NF− convergent in D 0 (in S 0 ) is not NF−convergent in D 0 (in S 0 ). This leads to inconsistency of the definitions of the product and convolution of distributions in the sense of the neutrix NF in D 0 and in S 0 . The net neutrices N F and N P are free from such incoherences. For example, if (τn )n∈N is an arbitrary numerical sequence such that τn → 0, then in the net (β τ )τ∈(0,1) of the form (3.15) of the neutrix N F one can find the corresponding sequence (βn ) of the form βn := β τn , i.e. of the form (3.14) with n replaced by τn−1 , while in the neutrix NF only the sequences (βn ) corresponding to the single sequence (τn ) of the form τn = n−1 are considered.

3.6 Neutrix powers of δ It will be convenient now to use the following notation for j ∈ N and, in particular, for j = 1:   Z j j D1 (R ) := ϕ ∈ D(R ) : ϕ(t) dt = 1 ; D1 := D1 (R1 ). (3.17) Rj

38

Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

We begin with giving the definitions of the product of k distributions in D 0 as well as the product and the Gaussian product of k tempered distributions in S 0 which are modifications and extensions of the sequential definition of the product of two distributions given in [1] (p. 242), [14] and [18]. Definition 3.5. Fix k ∈ N. For given f1 , . . . , fk ∈ D 0 ( f1 , . . . , fk ∈ S 0 ), the product f1 · . . . · fk in D 0 (in S 0 ) is defined by f1 · . . . · fk := lim ( f1 ∗ δτ ) · . . . · ( fk ∗ δτ ),

(3.18)

τ→0

if the above limit exists in D 0 (in S 0 ) for all delta-nets (δτ ) of the form δτ (x) = τ −1 σ (τ −1 x),

τ ∈ (0, 1), σ ∈ D1 , x ∈ R

(3.19)

and does not depend on σ , where D1 is defined in (3.17). In particular, if f ∈ D 0 ( f ∈ S 0 ), then formula (3.18) with f1 = . . . = fk = f defines the kth power f k of f in D 0 (in S 0 ). Definition 3.6. Let N be a neutrix in R(0,1) and k ∈ N. For given f1 , . . . , fk ∈ D 0 ( f1 , . . . , fk ∈ S 0 ), the N−product f1 · . . . · fk in D 0 (in S 0 ) is defined by f1 · . . . · fk := N − lim ( f1 ∗ δτ ) · . . . · ( fk ∗ δτ ),

(3.20)

τ→0

if the N−limit on the right hand side exists in D 0 (in S 0 ) for all delta-nets (δτ ) given by (3.19) and does not depend on σ . In particular, if f ∈ D 0 ( f ∈ S 0 ), then formula (3.20) with f1 = . . . = fk = f defines the kth N−power in D 0 (in S 0 ). Definition 3.7. For f1 , . . . , fk ∈ S 0 , the Gaussian product and the Gaussian N−product f1 · . . . · fk in D 0 (in S 0 ) is defined by (3.18) and by (3.20), respectively, whenever the limits in (3.18) and (3.20) exist in D 0 (in S 0 ) for all (δτ ) of the form (3.19), where σ is replaced by the single σ0 ∈ S given by 2

σ0 (x) := π −1/2 e−x ,

x ∈ R.

(3.21)

In particular, if f ∈ S 0 , then formulas (3.18) and (3.20) with f1 = . . . = fk = f (and σ = σ0 with σ0 given by (3.21)) define the kth Gaussian power and the kth Gaussian N−power f k of f , respectively, in D 0 (in S 0 ). Remark 3.5. Clearly, delta-nets (δτ ) and the net limits in (3.18) and (3.20) as τ → 0 can be equivalently replaced by delta-sequences (δn ) of the form: δn (x) = τn σ (τn x),

(τn ) ∈ d∞ , σ ∈ D1 , x ∈ R,

(3.22)

3. Convolution operators in the theory of circuits

39

and by the sequential limits as n → ∞, respectively. The class ∆m of all (δn ) of the form (3.22) called model delta-sequences was introduced in [14]. The product in D 0 given by (3.18) for k = 2 with (δn ) instead of (δτ ) and the respective sequential limit was studied first in [19] and then in [1] for other classes of delta-sequences. If the sequential limit exists in D 0 (in S 0 ) for all (δn ) ∈ ∆m , then it does not depend on (τn ) ∈ d∞ , but it may depend on σ ∈ D1 , as noticed in [15]. The same concerns the neutrix product of distributions in D 0 (in S 0 ). The additional assumption in Definitions 3.5 and 3.6 is made just to avoid such a dependence. Fisher and his followers in their papers on (neutrix) products of distributions [8, 9] use delta-sequences (δn ) of the form (3.22) with one fixed σ ∈ D1 (satisfying additional conditions) and one fixed (τn ), τn := n (n ∈ N), so their definition of the (neutrix) product of distributions depends, in general, on these particularly fixed σ and (τn ). Koh and Li [17] use delta-sequences (δn ) of the form (3.22) with the fixed σ0 of the form (3.21) instead of σ ∈ D1 and with the √ fixed (τn ) ∈ d∞ , this time given by τn := n (n ∈ N). The appearance of the two different particular (τn ) motivates additionally the use of arbitrary (τn ) ∈ d∞ in equation (3.22) or of delta-nets of the form (3.19). The product δ · δ and, more generally, the kth power δ k of δ for k ≥ 2 do not make sense in the standard approach [21] to the theory of distributions and do not exist in the sense of the Mikusi´nski product of distributions (see [1], pp. 243-244). However Koh and Li proved in [17] that δ k (k ≥ 2) exists in D 0 in the sense of Definition 3.7 of the Gaussian N−product (3.20) for f1 = . . . = fk = δ and N = NF . More exactly, they proved that the Gaussian NF−power δ k exists in D 0 for (δn ) of the form (3.22) with σ = σ0 given by (3.21) and particular √ τn := n. We extend below the result of Koh and Li replacing the neutrix N = NF of sequences by the neutrix N = N P (in particular N = N F ) of nets and the limit in D 0 by the stronger limit in S 0 : Theorem 3.1. The kth Gaussian N P -power (and the more N F−power) of δ exists in S 0 for arbitrary k ∈ N and the following formulas hold: δ2j = 0

( j ∈ N),

δ 2 j+1 =

1 (4π) j (2 j + 1) j+1/2 j!

δ (2 j)

( j ∈ N0 ). (3.23)

Proof. Fix ψ ∈ S and a delta-net (δτ ) of the form (3.19) with σ = σ0 given by equation (3.21), i.e. 2

δτ (x) := τ −1 σ0 (τ −1 x) = (τ 2 π)−1/2 e−(x/τ) ,

τ ∈ (0, 1), x ∈ R.

40

Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

Hence (δτk , ψ) = (τ 2 π)−k/2

Z

2

e−k(x/τ) ψ(x) dx,

ψ ∈S.

(3.24)

R

By Taylor’s formula and equation (3.24), there exists a certain ξ ∈ (0, 1) such that k−1

ψ (i) (0) R −k(x/τ)2 i e x i! i=0 R

(δτk , ψ) = (τ 2 π)−k/2 ∑ +

(τ 2 π)−k/2 R −k(x/τ)2 (k) e ψ (ξ x)xk k! R

dx + (3.25)

dx.

Putting t := (k/τ 2 )1/2 x we get Z

√ i+1 Z −t 2 i 2 e−k(x/τ) xi dx = τ/ k e t dt

R

i = 0, 1, . . . , k − 1

R

and Z

−k(x/τ)2

e

ψ

(k)

√ k+1 Z −t 2 (k) e ψ (ξ x)x dx = (τ/ k) k



 τξ t k √ t dt. k

R

R

Since ψ ∈ S (R), applying the Lebesgue’s dominated convergence theorem we have   Z Z 2 −t 2 (k) τξ t √ t k dt = ψ (k) (0) e−t t k dt, lim e ψ τ→0 k R

R

so the second addend in equation (3.25) tends to 0 as τ → 0. Hence, using (τ i+1−k )τ∈(0,1) as the elements of the neutrix N P for i = 0, 1, . . . , k − 2, we see from equation (3.25) that N P −lim (δτk , ψ) = τ→0

ψ (k−1) (0) (kπ)k/2 (k − 1)!

Z

2

e−t t k−1 dt

R

for arbitrarily fixed ψ ∈ S . Consequently, due to Definition 3.6, δ k exists in the sense of the kth Gaussian N P −power in S 0 and (−1)k−1 δ = δ (k−1) (kπ)k/2 (k − 1)! k

Z R

2

e−t t k−1 dt.

(3.26)

3. Convolution operators in the theory of circuits

41

Clearly Z

2

e−t t 2 j−1 dt = 0

Z

and

R

2

e−t t 2 j dt =

(2 j − 1)!! √ π, 2j

R

where (2 j − 1)!! := (2 j − 1) · (2 j − 3) · . . . · 3 · 1 for j ∈ N. Hence, by equation (3.26), formula (3.23) follows and the proof is completed. t u

3.7 Neutrix kth convolution of δ(k) and δ Definition 3.8. Let f ∈ D 0 (R) ( f ∈ S 0 (R)) and h ∈ D 0 (Rk ) (h ∈ S 0 (Rk )) for fixed k ∈ N. Let (δτ ) and (δeτ ) be delta-nets of the forms δτ (x) = τ −1 σ (τ −1 x),

τ ∈ (0, 1), σ ∈ D1 (R), x ∈ R

(3.27)

e ∈ D1 (Rk ), y ∈ Rk , τ ∈ (0, 1), σ

(3.28)

and δeτ (y) = τ −1 σe (τ −1 y),

respectively. Denote fτ := f ∗ δτ and hτ := h ∗ δeτ for τ ∈ (0, 1) and assume that k

the convolutions hτ ∗ fτ exist in D 0 (R) (in S 0 (R)) for arbitrary delta-nets (δτ ) and (δeτ ) of the forms (3.27) and (3.28), respectively, and for all τ ∈ (0, 1). k

The convolution of kth order h ∗ f in D 0 (R) (in S 0 (R)) is defined by k

k

h ∗ f := lim hτ ∗ fτ ,

(3.29)

τ→0

whenever the limit in (3.29) exists in D 0 (R) (in S 0 (R)) for arbitrary delta-nets e. (δτ ) and (δeτ ) of the form (3.27) and (3.28) and does not depend on σ or σ Definition 3.9. Let N be a neutrix in R(0,1) . Fix k ∈ N and let h and f be as in k

Definition 3.8. The N−convolution of kth order h ∗ f in D 0 (R) (in S 0 (R)) is defined by k

k

h ∗ f := N− lim hτ ∗ fτ ,

(3.30)

τ→0

whenever the N−limit in (3.30) exists in D 0 (R) (in S 0 (R)) for all delta-nets (δτ ) and (δeτ ) of the form (3.27) and (3.28), and does not depend on σ or σe . Definition 3.10. For h ∈ S 0 (Rk ) and f ∈ S 0 (R), the Gaussian convolution k

of kth order and the Gaussian N−convolution of kth order h ∗ f in D 0 (R) (in

42

Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek

S 0 (R)) are defined by (3.29) and by (3.30), respectively, whenever the limits in (3.29) and (3.30) exist in D 0 (R) (in S 0 (R)) for all delta-nets (δτ ) and (δeτ ) of the form (3.27) and (3.28), where σ and σe are replaced by σ0 ∈ S (R) and e0 ∈ S (Rk ), respectively, given by σ σ0 (x) := π −1/2 e−x

2

σe0 (y) := π −k/2 E(y),

and

2

2

for x ∈ R and y := (η1 , . . . , ηk ) ∈ Rk , where E(y) = e−η1 −...−ηk . Theorem 3.2. For k ∈ N, the Gaussian N P−convolution of kth order (and the more Gaussian N F−convolution of kth order) of the Dirac delta δ(k) in Rk and the Dirac delta δ in R exists in S 0 (R) and k

δ(k) ∗ δ = δ k

for

k ∈ N,

(3.31)

where δ k on the right hand side of (3.31) exists in the sense of the kth Gaussian N P -power (and the more N F−power) and is given by (3.23).

References [1] P. Antosik, J. Mikusi´nski and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier-PWN, Amsterdam-Warszawa, 1973. [2] E. Bedrosian and S. O. Rice, The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs, Proceedings of the IEEE 59 (1971), 1688-1707. [3] A. Borys, Consideration of Volterra series with excitation and/or impulse responses in the form of Dirac impulses, IEEE Transactions on Circuits and Systems II 57 (2010), 466-470. [4] A. Borys, A. Kami´nski and S. Sorek, Volterra systems and powers of Dirac delta impulses, Integral Transforms Spec. Func. 20 (2009), 301-308. [5] A. Borys and W. Sie´nko, Volterra series and multiplication of Dirac impulses, presentation by the first author at the conference ’Linear and Non-linear Theory of Generalized Functions’, Be¸dlewo, 2007. [6] J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North Holland, Amsterdam, New York, Oxford, 1984. [7] J. G. van der Corput, Introduction to the neutrix calculus, J. Analyse Math. 7 (1959/60), 281-398. [8] B. Fisher, The product of distributions, Quart. J. Math. Oxford 22 (2) (1971), 291298. −r (r−1) [9] B. Fisher, The neutrix distribution product x+ δ (x), Studia Sci. Math. Hungar. 9 (1974), 439-441. [10] N. J. Fliege, Systemtheorie, Teubner, Stuttgart, 1991. [11] E. G. Gilbert, Functional expansions for the response of nonlinear differential systems, IEEE Transactions on Automatic Control 22 (6) (1977), 909-921.

3. Convolution operators in the theory of circuits

43

[12] S. Haykin and B. Van Veen, Signals and Systems, Wiley, New York, 2003. [13] J. Horváth, Topological Vector Spaces and Distributions, Vol. 1, Addison-Wesley, Reading, 1966. [14] A. Kami´nski, Remarks on delta- and unit-sequences, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 26 (1978), 25-35. [15] A. Kami´nski, On model delta-sequences and the product of distributions, in Complex Analysis and Generalized Functions, I. Dimovski, V. Hristov, eds., Varna, 15–21 September 1991, 148-155. [16] A. Kami´nski and S. Sorek, Linear and nonlinear systems and neutrix powers of Dirac delta impulses, presentation by the second author at the 28th International Conference on Real Functions Theory, Stará Lesná, 2014. [17] E. L. Koh and C. K. Li, On the distributions δ k and (δ 0 )k , Math. Nachr. 157 (1992), 243-248. [18] J. Mikusi´nski, Criteria of the existence and of the associativity of the product of distributions, Studia Math. 21 (1962), 253-259. [19] J. Mikusi´nski, On the square of the Dirac delta-distribution, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 14 (1966), 511-513. [20] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Longman, New York, 1992. [21] L. Schwartz, Théorie des Distributions vols. I-II, Hermann, Paris, 1950-1951. [22] S. Sorek, On the notion of neutrix in the theory of generalized functions, in preparation.

A NDRZEJ B ORYS Faculty of Electrical Engineering, Gdynia Maritime University Morska 81-87, 81-225 Gdynia, Poland E-mail: [email protected]

´ A NDRZEJ K AMI NSKI Faculty of Mathematics and Natural Sciences, University of Rzeszów Prof. Pigonia 1, 35-310 Rzeszów, Poland E-mail: [email protected]

S ŁAWOMIR S OREK Faculty of Mathematics and Natural Sciences, University of Rzeszów Prof. Pigonia 1, 35-310 Rzeszów, Poland E-mail: [email protected]

Chapter 4

Continuity connected with ψ-density

MAŁGORZATA FILIPCZAK AND MAŁGORZATA TEREPETA

2010 Mathematics Subject Classification: 54A10, 28A05. Key words and phrases: density point, ψ-density point, density topology, ψ-density topology, approximately continuous functions, ψ-approximately continuous functions.

4.1 Introduction The notion of approximately continuous functions was introduced at the beginning of the XXth century. In 1915 in [5], A. Denjoy stated that a real valued function f is approximately continuous at a point x0 if and only if there exists a measurable set A ⊂ R such that lim

h→0+

m(A ∩ [x0 − h, x0 + h]) = 1 and f (x0 ) = lim f (x), x→x0 2h

(4.1)

x∈A

where m denotes the Lebesgue measure. The point x0 which fulfills the first from the above equalities is called a density point of a set A. Denjoy discovered that approximately continuous functions are of Baire class 1 and have Darboux property. He also proved that a real function defined on R is measurable if and only if it is approximately continuous almost everywhere on R. The first definition did not involve the concept of density topology, which appeared later in the paper [14]. In 1961 ([13]) C. Goffman and D. Waterman examined ap-

46

Małgorzata Filipczak and Małgorzata Terepeta

proximately continuous transformations from euclidean space into an arbitrary metric space. They defined the density topology (denoted by Td ) finer then the natural topology To and proved that the approximately continuous transformations are Darboux Baire 1 and are continuous as f : (R, Td ) → (R, To ). In the same year it was shown ([12]) that Td is the coarsest topology relative to which approximately continuous functions are continuous, because the density topology on R is completely regular (but not normal). By putting density topology on the domain and on the range of a function we can obtain another class of continuous functions: density continuous functions, which were deeply examined by K. Ciesielski, L. Larson and K. Ostaszewski (for instance [3], [4], [20]), J. Niewiarowski ([19]) and A. Bruckner ([1]). In 1910 ([16]) H. Lebesgue proved that for any Lebesgue measurable set A ⊂ R the equality m (A ∩ [x − h, x + h]) =1 (4.2) 2h holds for all points x ∈ A except for the set of Lebesgue measure zero. We denote by L the family of all Lebesgue measurable sets on R. Equivalently we can say that m (A∆ Φd (A)) = 0 for any A ∈ L (∆ stands for symmetric difference), where   m (A ∩ [x − h, x + h]) Φd (A) = x ∈ R : lim+ =1 . 2h h→0 lim

h→0+

The family Td = {A ∈ L : A ⊂ Φd (A)} forms a topology called the density topology. (R, Td ) is a Baire space and Td is invariant under translations and multiplications by nonzero numbers. The families of meager sets and nowhere dense sets in (R, Td ) coincide and both are equal to the family of Lebesgue null sets. Any set of positive inner measure has nonempty interior in Td . In 1959 ([21]) S. J. Taylor solved the problem presented by S. Ulam in The Scottish Book. Taylor’s results were the contribution to the development of ψdensity topology and another classes of continuous functions, with density and ψ-density on the domain and the range. Let us present his two main theorems. Theorem 4.1 ([21, Theorem 3]). For any Lebesgue measurable set A ⊂ R there exists a function ψ : (0, ∞) → (0, ∞) which is continuous, nondecreasing and lim+ ψ(x) = 0 such that

x→0

4. Continuity connected with ψ-density

47

m(A0 ∩ I) =0 m(I)→0 m(I)ψ(m(I)) lim

for almost all x ∈ A, where I is any interval containing x (A0 stands for the complement of A). Theorem 4.2 ([21, Theorem 4], compare with [22, Theorem 0.2]). For any function ψ : (0, ∞) → (0, ∞) which is continuous, nondecreasing, lim+ ψ(x) = x→0

0, and for any real number α ∈ (0, 1), there exists a perfect set E ⊂ [0, 1] such that m(E) = α and m(E 0 ∩ I) lim sup =∞ m(I)→0 m(I)ψ(m(I)) for all x ∈ E. Following Taylor we introduce a notion of ψ-density (compare with [22]). Let C be the family of all nondecreasing continuous functions ψ : (0, ∞) → (0, ∞) such that lim+ ψ(t) = 0. t→0

We say that x ∈ R is a ψ-density point of A ∈ L (we will write x ∈ Φψ (A)) if and only if m(A0 ∩ [x − h, x + h]) lim+ = 0. 2hψ(2h) h→0 From Theorem 4.2 we obtain that the operator Φψ is not a lower density operator. However, this operator is an almost lower density operator (see [23] and [15]) and the family Tψ = {A ∈ L : A ⊂ Φψ (A)} forms a topology called ψ-density topology. Clearly, To Tψ Td . Let us notice that, despite of topologies generated by lower density operators, for any ψ-density topology there is a set of positive measure and empty Tψ -interior. On the other hand, Tψ has a lot of properties similar to properties generated by lower density operators. For any ψ ∈ C null sets are Tψ -closed and consequently the space (R, Tψ ) is neither first countable, nor second countable, nor Lindelöf, nor separable. A set is compact in Tψ if and only if it is finite; a set is connected if and only if it is connected in To ; a set is Tψ -Borel if and only if it is measurable (for details see [11]). For any topologies Ta , Tb ⊂ 2R we will denote by Cab be the family of all continuous functions f : (R, Ta ) → (R, Tb ). It is easy to observe that (P1) for any topologies Ta , Tb and Tc , if Tb ⊂ Tc then Cab ⊃ Cac and Cba ⊂ Cca ;

48

Małgorzata Filipczak and Małgorzata Terepeta

(P2) for any pair of topologies Ta , Tb , if Ta ⊂ Tb then Cab ⊂ Caa ∩ Cbb and Cba ⊃ Caa ∪ Cbb . If we start with topologies considered above: natural topology To , density topology Td and ψ-density topology Tψ , we may obtain nine classes of continuous functions: Coo , Cod , Coψ , Cdo , Cdd , Cdψ , Cψo , Cψd and Cψψ .

Fig. 1. Relations between the classes of continuous functions connected with To , Td , Tψ

Denote by F the set of all these nine families. Some of them have been examined: 1. Coo is the class of all ordinary continuous functions. 2. Cdo is the class of approximately continuous functions (see [4], [12] and [23]). Let us remind that such functions are Darboux Baire 1 (a function f is Baire 1 if for any perfect set P the restriction f |P has a point of continuity). Moreover, each approximately continuous and bounded function is a derivative. 3. Cdd is the class of density continuous functions (for the results see [2], [4] and [20]). This class is not additive. All density continuous functions are approximately continuous and they belong to Darboux Baire∗ 1 class. Remind that f is Baire∗ 1 if for each perfect set P there is a portion Q ⊂ P such that f |Q is continuous.

4. Continuity connected with ψ-density

49

We will consider properties of functions from classes connected with ψdensity topology. We present selected results of published articles and some open problems. For the convenience of the reader we quote examples and sketches of the proofs. Observe that all the families from F have some common properties. Assume that Cab ∈ F . It is evident that 1. C onst ⊂ Cab , 2. if f ∈ Cab , then − f ∈ Cab , 3. if f ∈ Cab and k ∈ R, then f + k ∈ Cab . Moreover, Cab forms a lattice. Remark 4.1. If f , g ∈ Cab then max{ f , g} ∈ Cab and min{ f , g} ∈ Cab . Proof. Let h = max{ f , g} and x0 ∈ R. We will show that the function h : (R, Ta ) → (R, Tb ) is continuous at x0 . If f (x0 ) > g(x0 ) (or g(x0 ) > f (x0 )) then h is equal to f (g) on some neighbourhood of x0 and the thesis is obvious. Assume then that h(x0 ) = f (x0 ) = g(x0 ) and G ∈ Tb is an open neighbourhood of the point h(x0 ). The sets f −1 (G) and g−1 (G) are open and x0 ∈ H = f −1 (G) ∩ g−1 (G) ∈ Ta . As min{ f , g} = max{− f , −g} we obtain that min{ f , g} ∈ Cab .

t u

If f , g ∈ Cab then f + g and k f for a real k may not be in Cab . Also the limit of uniformly convergent sequence of functions from Cab may not belong to Cab .

4.2 Continuity related to natural topology It appears that classes Coψ and Cod are surprisingly small. Following [4, Theorem 3.1] and [8, Theorem 3] we can check that: Theorem 4.3. Coψ = C onst for any function ψ ∈ C . Indeed, suppose that f ∈ Coψ and a < b. Then f ([a, b]) is compact and connected in Tψ . Therefore, f ([a, b]) is a singleton and f has to be constant. Another proof can be found in [8]. By (P1) we obtain that Cod ⊂ Coψ , hence Cod = C onst. Let us examine classes Cψo . First we introduce the notion of the inner ψdensity point of a set A ⊂ R: x is said the inner ψ-density point of A ⊂ R if

50

Małgorzata Filipczak and Małgorzata Terepeta

and only if there exists a set B ∈ L such that B ⊂ A and x is a ψ-density point of B. Clearly, for a measurable set the notions of ψ-density point and inner ψdensity point are equivalent. Now we can define ψ-approximately continuous function. Definition 4.1 ([22]). We say that a function f : R → R is ψ-approximately continuous at x0 if and only if x0 is the inner ψ-density point of f −1 (( f (x0 ) − ε, f (x0 ) + ε)) for each ε > 0. It is evident, that f is ψ-approximately continuous at x0 if and only if x0 is the inner ψ-density point of f −1 ((a, b)) for each interval (a, b) such that f (x0 ) ∈ (a, b). We say that a function f : R → R is ψ-approximately continuous if and only if f is ψ-approximately continuous at each point. Note that a set A is open in the topology Tψ if and only if each point of A is the inner ψ-density point of A ([22, Theorem 3.3]). Therefore, a function f : R → R is ψ-approximately continuous if and only if for each interval (a, b) the set f −1 ((a, b)) belongs to Tψ ([22, Theorem 3.6]), so f ∈ Cψo . As To ⊂ Tψ ⊂ Td we obtain Coo ⊂ Cψo ⊂ Cdo . From [7, Property 5] it is known that for any function ψ ∈ C there are functions ψ1 , ψ2 ∈ C such that Cψ1 o Cψo Cψ2 o . Hence for any function ψ ∈ C Coo

Cψo

Cdo

DB 1 .

Moreover, in [24, Theorem 9] it is proved that the family Cψo is not contained in the class of Baire∗ 1 functions. It is easy to check that Theorem 4.4. Let ψ ∈ C . 1. If f , g are ψ-approximately continuous functions, then f + g, f − g, f · g, max{ f , g} and min{ f , g} are ψ-approximately continuous. If f is a ψ-approximately continuous function and f (x) 6= 0 for x ∈ X, then 1f is ψ-approximately continuous. 2. If fn ∈ Cψo for any n ∈ N and a sequence { fn }n∈N uniformly converges to f , then f ∈ Cψo . 3. If f ∈ Cψo and g ∈ Coo , then g ◦ f ∈ Cψo .

4. Continuity connected with ψ-density

51

Remind that a function f : R → R is called measurable if f −1 (U) ∈ L for any U ∈ To . Obviously, any approximately continuous and any ψ-approximately continuous function are measurable. The famous theorem of Denjoy ([5]) states that f is measurable if and only if it is approximately continuous almost everywhere. The similar theorem holds for ψ-approximate continuity. Theorem 4.5 ([22, Theorem 3.7]). A function f : R → R is measurable if and only if there exists a function ψ ∈ C such that f is ψ-approximately continuous almost everywhere. Proof. In the face of the theorem of Denjoy it is sufficient to prove that if f is measurable, then there exists ψ ∈ C such that f is ψ-approximately continuous almost everywhere. Suppose that f is measurable. Let (an , bn ), n ∈ N, be a basis of Euclidean topology on the real line. From [22, Theorem 2.12] there exists a function ψ ∈ C such that for any n ∈ N almost all points of f −1 ((an , bn )) are its ψ-density points. Denote by N( f ) the set of all points in which f is not ψ-approximately continuous. Then N( f ) ⊂

[

[ f −1 ((an , bn )) \ Φψ ( f −1 ((an , bn )))]

n∈N

t u

and consequently m(N( f )) = 0.

It is known that if Tψ1 \ Tψ2 6= 0, / then Cψ1 o \ Cψ2 o 6= 0/ ([7, Theorem 2]). From [7, Proposition 4] it follows that there exist continuum different topologies Tψb Tψa , where 0 < a < b < 1, and continuum different classes of continuous functions such that Coo

Cψb o

Cψa o

Cdo

for any 0 < a < b < 1. Moreover, if Cψb o Cψa o , then there exists a c-generated algebra F of functions which is contained in the difference Cψa o Cψb o (compare [17]).

4.2.1 Classes Cψψ Let us fix a function ψ from the family C . We will consider continuous functions f : (R, Tψ ) → (R, Tψ ). Such functions are called ψ-continuous. Evidently, C onst Cψψ ⊂ Cψo Cdo .

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Małgorzata Filipczak and Małgorzata Terepeta

Since Cdo DB 1 ([4, Theorem 1.3.1]) any function from Cψψ is measurable and DB 1 . Remind some additional information about ψ-density topologies. All of them are invariant under translation, but they may not be invariant under multiplication. More precisely, if |α| ≥ 1 and 0 is a ψ-density point of measurable set A, then 0 is a ψ-density point of αA. Indeed, it follows from monotonicity of ψ ∈ C and inequality m(αA0 ∩ [−h, h]) αm(A0 ∩ [− αh , αh ]) αm(A0 ∩ [− αh , αh ]) = ≤ . 2h 2h 2h 2hψ(2h) α ψ(2h) α ψ( α ) On the other hand, if lim sup ψ(αx) ψ(x) = ∞, then there exists a set A ∈ Tψ such x→0+

that α1 A ∈ / Tψ (compare with [22, Theorem 2.8]). It is not difficult to check (compare [6] and [22]), that the topology Tψ is invariant under multiplication by a nonzero number if and only if ψ fulfills the condition lim sup x→0+

ψ(2x) < ∞. ψ(x)

(∆2 )

We will write then ψ ∈ ∆2 . Proposition 4.1 ([8, Remark 9]). Assume that ψ ∈ C . (1) If ψ ∈ ∆2 and f ∈ Cψψ , then k f ∈ Cψψ for any number k ∈ R. (2) If ψ ∈ ∆2 , then any piecewise linear function is ψ-continuous. (3) If ψ ∈ / ∆2 , then no linear function f (x) = kx with |k| > 1 is ψ-continuous. For ψ = id the topology Tψ coincides with superdensity topology ([18]). This function evidently fulfills (∆2 ). The functions ψ(x) = xα for α ≥ 1 are the most useful for obtaining topologies Tψ satisfying (∆2 ). Note that, there are 2 functions which do not satisfy (∆2 ) but are in C , for instance ψ(x) = e− ln x for x ∈ (0, 1) and linear for x ≥ 1 (one can find another example in [6]). Let us remind that if ψ ∈ / ∆2 then even linear functions may not be continuous. In [10] it is proved that if ψ ∈ ∆2 and there exist numbers α, β > 0 such that f (x) − f (y) 0. Then there is a nonempty set U ⊂ f (A) open in topology Td . Since f is continuous, f −1 (U) ∈ Tψ . But A0 is dense in Tψ , then f −1 (U) ∩ A0 6= 0/ and, consequently, U ∩ f (A0 ) 6= 0. / It is a contradiction, because U ⊂ f (A). This proves that m( f (A)) = 0.

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Małgorzata Filipczak and Małgorzata Terepeta

Consequently, the set f (A) and any subset B of f (A) is closed in Td . Since f ∈ Cψd , f −1 (B) is closed in Tψ for any B ⊂ f (A). Any Tψ -closed set is measurable. Obviously, for any C ⊂ A there exists B ⊂ f (A) such that f −1 (B) = C. It gives a contradiction, because the set A (of positive measure) contains a nonmeasurable set. t u Modifying the last proof we can prove the following Theorem 4.10 (compare [8]). If f ∈ Cψd and a < b then f |(a,b) is not an injection. We do not know if there exists nonconstant function in Cψd . From (P2) it follows that Cdψ ⊃ Cdd ∪ Cψψ and Cdψ ⊂ Cdo , therefore functions from Cdψ are DB 1 and may not be DB ∗1 . The inclusion Cdψ ⊂ Cdo is proper. Indeed, take the interval set A = put

∞ S

[an , bn ] such that 0 ∈ Φ + (A) and

n=1

1 for x ∈ [an , bn ], n ∈ N, n f (x) = linear for x ∈ [bn , an−1 ] , n ∈ N  0 for x ≤ 0. / Td . Hence f ∈ / Then Y = (0, ∞) \ { 1n ; n ∈ N} ∈ Tψ , but f −1 (Y ) = (0, ∞) \ A ∈ Cψd . Simultaneously, f is continuous, therefore approximately continuous, so Cdψ Cdo and, additionally, we obtained that Coo 6⊂ Cψd . S Cdψ 6= 0. / Moreover f ∈ Cdo \ Cdψ for any ψ ∈ C . Therefore, Cdo \ ψ∈C

However we do not know how to describe the union

S

Cdψ . Obviously, if

ψ∈C

Tψ1 Tψ2 then Cdψ2 ⊂ Cdψ1 , but we even can not say whether this inclusion is proper.

4.3 Functions preserving ψ-density points In [1] there was introduced the concept of homeomorphism preserving density points. This notion was examined also in [19]. We will adopt this idea to the theory of ψ-density continuous functions. Fix a function ψ ∈ C and let introduce the notion of a function preserving ψ-density points. Definition 4.2. We will say that a homeomorphism h preserves ψ-density points if for any measurable set S ⊂ R and any x0 ∈ Φψ (S)

4. Continuity connected with ψ-density

lim+

t→0

57

m∗ ((h(S))0 ∩ [h(x0 ) − t, h(x0 ) + t]) =0 2tψ(2t)

(m∗ stands for the outer Lebesgue measure). Observe that if a homeomorphism h preserves ψ-density points, then it also preserves ψ-dispersion points. Proposition 4.3. If h is a homeomorphism preserving ψ-density points, then h satisfies Lusin’s condition (N). Proof. Let Z be a set of Lebesgue measure zero. There exists a Gδ -set A ⊃ Z of measure zero. Then h(A) is also a Gδ -set, so it is measurable. Suppose that Lebesgue measure of h(A) is positive. Hence h(A) has density 1 at a certain point y0 ∈ h(A): m(h(A) ∩ [y0 − t, y0 + t]) = 1. lim+ 2t t→0 Observe that for S = A0 and any t > 0 such that ψ(2t) ≤ 1, we have m∗ ((h(S))0 ∩ [y0 − t, y0 + t]) m∗ (h(S0 ) ∩ [y0 − t, y0 + t]) = ≥ 2tψ(2t) 2tψ(2t) m(h(A) ∩ [y0 − t, y0 + t]) ≥ , 2t therefore h does not preserve ψ-density points.

t u

Corollary 4.1. If a homeomorphism h : [0, 1] → [0, 1] preserves ψ-density points then it is an absolutely continuous function. From Proposition 4.3 it follows that if the homeomorphism h preserves ψdensity points then, for any measurable set S ⊂ R, h(S) is a measurable set and we need not use the outer measure in Definition 4.2. Theorem 4.11. A homeomorphism h : R → R preserves ψ-density points if and only if h−1 is a ψ-continuous function. Proof. First we assume that h preserves ψ-density points. We will show that h−1 is a ψ-continuous function at any point. Fix a point y0 and a set V ∈ Tψ such that x0 = h−1 (y0 ) ∈ V . We will show that there exists a set U ∈ Tψ such that y0 ∈ U and h−1 (U) ⊂ V . Since V is open in Tψ , for any x ∈ V we have x ∈ Φψ (V ). The homeomorphism preserves ψ-density points, so h(x) ∈ Φψ (h(V )). Hence h(V ) is open in Tψ and putting U = h(V ), we complete the proof of this implication.

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Małgorzata Filipczak and Małgorzata Terepeta

Suppose now that h does not preserve ψ-density points. Set x0 ∈ R and S ∈ L such that x0 is a ψ-dispersion point of S and lim sup t→0+

m∗ (h(S) ∩ [h(x0 ) − t, h(x0 ) + t]) > 0. 2tψ(2t)

Take a Gδ -set A ⊃ S such that m(A \ S) = 0, a sequence (an )n∈N decreasing to 0 and a number α > 0 for which m (h(A) ∩ [h(x0 ) − an , h(x0 ) + an ]) >α 2an ψ(2an ) for all n ∈ N. We can assume that for n ∈ N m (h(A) ∩ [h(x0 ), h(x0 ) + an ]) α > 2an ψ(2an ) 2

(4.3)

For any natural n there exists a closed set Bn ⊂ h(A) ∩ [h(x0 ) + an+1 , h(x0 ) + an ] such that   α 1 m(Bn ) > m h(A) ∩ [h(x0 ) + an+1 , h(x0 ) + an ] − · n · 2an ψ(2an ). (4.4) 4 2 The set B =

∞ S

Bn ∪ {h(x0 )} is closed in natural topology and from (4.3) and

n=1

(4.4) we obtain m (B ∩ [h(x0 ), h(x0 ) + an ]) α ≥ >0 2an ψ(2an ) 4 for any n. Hence h(x0 ) is not a ψ-dispersion point of the set B. On the other hand, x0 is a ψ-dispersion point of h−1 (B) and the set C = R \ h−1 (B) ∪ {x0 } ∈ Tψ , but h(C) = R \ B ∪ {h(x0 )} ∈ / Tψ , so the function h−1 is not ψ-continuous. t u This survey still leaves numerous questions without answers. In particular we do not know if: 1. Is the difference Cψψ \ Coo nonempty and is the class Cψψ closed under uniform convergence if ψ ∈ / ∆2 ? 2. Does there exist a nonconstant function f ∈ Cψd ? 3. Is the inclusion Cdψ2 ⊂ Cdψ1 proper, if Tψ1 Tψ2 ? S T 4. What is the union Cdψ and intersection Cdψ ? ψ∈C

ψ∈C

5. What is the relation between classes Cψ1 ψ1 and Cψ2 ψ2 for different functions ψ1 , ψ2 ∈ C such that Tψ1 6= Tψ2 ?

4. Continuity connected with ψ-density

59

References [1] A. M. Bruckner, Density-preserving homeomorphisms and a theorem of Maximoff, Quart. J. Math. Oxford 21 (1970), 337-347. [2] M. Burke, Some remarks on density-continuity functions, Real Anal. Exch. 14 (1988/89) no. 1, 235-242. [3] K. Ciesielski, L. Larson, Various continuities with the density, I-density and ordinary topology on R, Real Anal. Exch. 17 (1991/92), 183-210. [4] K. Ciesielski, L. Larson, K. Ostaszewski, I-density continuous functions, Memoirs of the Amer. Math. Soc. 107 (1994). [5] A. Denjoy, Sur les fonctions sérivées sommables, Bull. Soc. Math. France 43 (1915), 161-248. [6] M. Filipczak, σ -ideals, topologies and multiplication, Bull. Soc. des Sci., Łód´z 2002, Vol. LII, 11-16. [7] M. Filipczak, Families of ψ-approximate continuous functions, Tatra Mt. Math. Publ. 28 (2004), 219-225. [8] M. Filipczak, M. Terepeta, On continuity concerned with ψ-density topologies, Tatra Mt. Math. Publ. 34 (2006), 29-36. [9] M. Filipczak, M. Terepeta, ψ-continuous functions, Rend. Circ. Mat. Palermo 58(2) (2009), 245-255. [10] M. Filipczak, M. Terepeta, ψ-continuous functions and functions preserving ψdensity points, Tatra Mt. Math. Publ. 42 (2009), 1-12. [11] M. Filipczak, M. Terepeta, On ψ-density topologies on the real line and on the plane, Traditional and present-day topics in real analysis, Faculty of Mathematics and Computer Science. University of Łód´z, Łód´z 2013, 367-387. [12] C. Goffman, C. J. Neugebauer, T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497-506. [13] C. Goffman, D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 161-121. [14] O. Haupt, C. Pauc, La topologie approximative de Denjoy envisagée vraie topologic, C. R. Acad. Sci. Paris 234 (1952), 390-392. [15] J. Hejduk, R. Wiertelak, On the abstract density topologies generated by lower and almost lower density operators, Traditional and present-day topics in real analysis, Faculty of Mathematics and Computer Science. University of Łód´z, Łód´z 2013, 431447. [16] H. Lebesgue, Sur l’intégration des fonctions discontinues, Ann. Sci. Éc. Norm. Supér. 27 (1910), 361-450. [17] S. Lindner, M. Terepeta, Algebrability within the class of Baire 1 functions, accepted for publishing in Lith. Math. J. [18] J. Lukes, J. Maly, L. Zajicek, Fine Topology Methods in Real Analysis and Potential Theory, Springer-Verlag 1189, 1986. [19] J. Niewiarowski, Density preserving homeomorphisms, Fund. Math. 106 (1980), 7787. [20] K. Ostaszewski, Continuity in the density topology, Real Anal. Exch. 7 (1981/82) no. 2, 259-270. [21] S. J. Taylor, On strengthening of the Lebesgue Density Theorem, Fund. Math. 3.46 (1959), 305–315.

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[22] M. Terepeta, E. Wagner-Bojakowska, ψ-density topologies, Circ. Mat. Palermo 48.3 (1999), 451–476. [23] W. Wilczy´nski, Density Topologies, Handbook of Measure Theory, North-Holland, Amsterdam (2002), 675–702. [24] E. Wagner-Bojakowska, Remarks on ψ-density topology, Atti Sem. Mat. Fis. Univ. Modena, IL, (2001), 79–87.

M AŁGORZATA F ILIPCZAK Department of Mathematics and Computer Science, Łód´z University ul. S. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

M AŁGORZATA T EREPETA Center of Mathematics and Physics, Łód´z University of Technology al. Politechniki 11, 90-924 Łód´z, Poland E-mail: [email protected]

Chapter 5

On equivalence of topological and restrictional continuity

KATARZYNA FLAK AND JACEK HEJDUK

2010 Mathematics Subject Classification: 54A05, 28A05. Key words and phrases: lower density operator, topological continuity, restrictional continuity.

5.1 Introduction Let R denote the set of reals and N the set of positive integers. By τ0 we shall denote the natural topology on R. Let B(τ), K(τ), Ba(τ) denote the family of all Borel sets, meager sets and sets having the Baire property in a topological space (R, τ), respectively. A τ-open set A ⊂ R is τ-regular if A = intτ clτ A , where intτ and clτ mean the interior and closure with respect to the topology τ. If τ = τ0 then we shall use the notation B, K and Ba, respectively. The symmetric difference of sets A, B is denoted by A M B. Let Φ : τ0 → 2R be an operator satisfying the following conditions: (i) Φ(0) / = 0, / Φ(R) = R, (ii) ∀ ∀ Φ(A ∩ B) = Φ(A) ∩ Φ(B), A∈τ0 B∈τ0

(iii)

∀ A ⊂ Φ(A).

A∈τ0

Let Φ stand for the family for all operators satisfying conditions (i) − (iii). Remark 5.1. If Φ ∈ Φ then Φ(A) ⊂ clτ0 A for every A ∈ τ0 .

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It is well known that every set A ∈ Ba has the unique representation A = G(A) M B where G(A) is a regular open set and B ∈ K (cf. [4]). In particular, if V ∈ τ0 then V = W \ P where W is regular open and P is a nowhere dense closed set (see [5]). Let Φ ∈ Φ and Φr : Ba → 2R be defined by formula ∀ Φr (A) = Φ(G(A)).

A∈Ba

The following theorems are a special case of similar theorems in [1] concerning arbitrary topological Baire spaces. Theorem 5.1. For every Φ ∈ Φ, the operator Φr is a lower density operator on (R, Ba, K). This means that the following conditions are satisfied: 1◦ Φr (0) / = 0, / Φr (R) = R, ◦ 2 ∀ ∀ Φr (A ∩ B) = Φr (A) ∩ Φr (B), 3◦ 4◦

A∈Ba B∈Ba



∀ A M B ∈ K ⇒ Φr (A) = Φr (B),

A∈Ba B∈Ba

∀ A M Φr (A) ∈ K.

A∈Ba

Theorem 5.2. For every operator Φ ∈ Φ, the family TΦr = {A ∈ Ba : A ⊂ Φr (A)} is a topology on R strictly stronger than τ0 . Proof. Since the pair (Ba, K) has the hull property, what means that every family of pairwise disjont sets having the Baire property but not meager is at most countable, and Φr is a lower density operator on (R, Ba, K), we infer that the family TΦr = {A ∈ Ba; A ⊂ Φr (A)} is a topology on R, called an abstract density topology on (R, Ba, K) (see [4], p. 208 and p. 213). If V ∈ τ0 then by Remark 5.1, V = W \ P where W is a regular open set and P ∈ K. Hence G(A) = W and Φr (V ) = Φ(W ) ⊃ W ⊃ V . Therefore V ∈ TΦr . Evidently, the set of irrational numbers is a member of TΦr \ τ0 , so the proof is complete. t u The next theorem lists properties of the topological space (R, TΦr ). For the proofs and some related comments see Theorem 4 in [1]. Theorem 5.3. Let Φ ∈ Φ. Then the topological space (R, TΦr ) has the following properties: a) A ∈ K iff A is TΦr -nowhere dense and closed, b) K(TΦr ) = K,

5. On equivalence of topological and restrictional continuity

63

c) Ba(TΦr ) = B(TΦr ) = Ba, d) (R, TΦr ) is the Baire space, e) A ⊂ X is compact iff A is finite, f ) (R, TΦr ) is neither separable, nor first countable or second countable, g) (R, TΦr ) is not a Lindelöf space, h) if A ⊂ R then IntΦr (A) = A ∩ Φr (B), where B ∈ Ba is a kernel of A. Some examples of operators belonging to Φ have already been considered in the literature. Example 5.1. Let Φ = Φd , where Φd denotes the density operator on the family of Lebesgue measurable sets in R. Then Φ ∈ Φ; the topology TΦr = {A ∈ Ba : A ⊂ Φr (A)} was intensively investigated in [11] and some generalization of this approach is presented in [10]. Example 5.2. Let Φ = ΦΨ , where ΦΨ denote the Ψ -density operator on the family of Lebesgue measurable sets in R (see [11]). Then Φ ∈ Φ; the topology TΦr = {A ∈ Ba : A ⊂ ΦΨ (A)} was investigated in [8]. Example 5.3. Let Φ(A) = A for every A ∈ τ0 . Then Φ ∈ Φ and TΦr = {B ⊂ R : B = C \ D, C ∈ τ0 , D ∈ K}, (see in [1] and [3]). Example 5.4. Let Φ = ΦI , where ΦI denote the I -density operator on the family Ba in R (see [5]). Then Φ ∈ Φ and for every set A ∈ Ba, Φr (A) = Φ(G(A)) = Φ(A). This implies that TΦr = TI , where TI is the I -density topology (see [6]).

5.2 The main results In the following part we shall focus on two kinds of continuity: topological and restrictional. Let Φ ∈ Φ. Definition 5.1. A function f : R → R is TΦr -topologically continuous at x0 ∈ R if ∀ ∃ (x0 ∈ A ∧ A ⊂ {x : | f (x) − f (x0 )| < ε}). ε>0 A∈TΦr

Obviously, a function f : X → R is TΦr -topologically continuous at every point x ∈ X if and only if it is continuous as a transformation from the topological space (X, TΦr ) to (R, τ0 ).

64

Katarzyna Flak and Jacek Hejduk

Definition 5.2. We shall say that a function f : R → R is TΦr -restrictionally continuous at x0 ∈ R if there exists a set E ∈ Ba such that x0 ∈ Φr (E) and f |E is τ0 -continuous at x0 . Property 5.1. (cf. [1]) Let Φ ∈ Φ. If f : R → R is TΦr -restrictionally continuous at x0 ∈ R then f is TΦr -topologically continuous at x0 . Proof. Assume that f is TΦr -restricionally continuous at x0 ∈ R. Then there exists a set E ∈ Ba such that x0 ∈ Φr (E) and f |E is τ0 - continuous at x0 . Thus, for every ε > 0 there exist V ∈ τ0 such that x0 ∈ V and E ∩ V ⊂ {x ∈ R : | f (x) − f (x0 )| < ε}. Then x0 ∈ A = E ∩ Φr (E) ∩ V ∈ TΦr and A ⊂ {x ∈ R : | f (x) − f (x0 )| < ε}. This means that f is TΦr -topologically continuous at x0 . t u The converse is not true. Namely, if Φ = ΦI then TΦr = TI , and it was proved in [6] that TI -topological continuity and TI -restrictional continuity are not equivalent.It is also worth mentioning that the topologies in papers [12] and [9] are such that topological and restrictional continuity are not equivalent. However, if Φ = Φd or Φ = ΦΨ , the paper [8] contains the proof of equivalence of both kinds of continuity. By Corollary 3 in [1] we obtain the following theorem giving equivalence of topological and restrictional continuity on residual sets. Theorem 5.4. Let Φ ∈ Φ and f : R → R. If C1 ( f ) and C2 ( f ) are the sets of TΦr -topological continuity and TΦr - restrictional continuity respectively, then C1 ( f ) is residual if and only if C2 ( f ) is residual with respect to topology τ0 . Now, we characterize the equivalence of topological and restrictional continuity in terms of the TΦr -topology for every Φ ∈ Φ. Theorem 5.5. Let f : R → R, Φ ∈ Φ and x0 ∈ R. The following conditions are equivalent: (a) f is TΦr -topologically continuous at x0 if and only if f is TΦr -restrictionally continuous at x0 ; T (b) for every decreasing sequence {En }n∈N ⊂ Ba such that x0 ∈ ∞ n=1 Φr (En ) there exists a sequence {rn }n∈N ⊂ R+ with rn & 0 such that S x0 ∈ Φr ( ∞ n=1 En ∩ (R \ (x0 − rn , x0 + rn ))); T (c) for every decreasing sequence {En }n∈N ⊂ τ0 such that x0 ∈ ∞ n=1 Φr (En ) there exists a sequence {rn }n∈N ⊂ R+ with rn & 0 such that S x0 ∈ Φr ( ∞ n=1 (En ∩ (R \ (x0 − rn , x0 + rn ))));

5. On equivalence of topological and restrictional continuity

65

(d) for every decreasing sequence {En }n∈N of τ0 -regular open sets such that T x0 ∈ ∞ (En ) there exists a sequence {rn }n∈N ⊂ R+ with rn & 0 such n=1 Φr S that x0 ∈ Φr ( ∞ n=1 (En ∩ (R \ (x0 − rn , x0 + rn )))). Proof. By Theorem 4 in [2] (see also Theorem 3.1 in [7]) conditions (a) and (b) are equivalent. Obviously, (b) ⇒ (c) ⇒ (d). We shall prove (d) ⇒ (b). T Let {En }n∈N ⊂ Ba be a decreasing sequence such that x0 ∈ ∞ n=1 Φr (En ). Then {G(En )}n∈N is a decreasing sequence of regular open sets such that T Φr (En ) = Φr (G(En )) for all n ∈ N, and x0 ∈ ∞ there exn=1 Φr (G(En )). Then S∞ ists a sequence {rn }n∈N ⊂ R+ with rn & 0 such that x0 ∈ Φr ( n=1 (G(En ) ∩ S (R \ (x0 − rn , x0 + rn )))) = Φr ( ∞ t u n=1 (En ∩ (R \ (x0 − rn , x0 + rn )))). Property 5.2. If Φ(A) = A for every A ∈ τ0 , then Φ ∈ Φ and for every function f : R → R , TΦr -topological continuity and TΦr -restrictional continuity are equivalent. Proof. Evidently Φ ∈ Φ. It is sufficient to prove condition (a) of Theorem 5. Let {En }n∈N be a decreasing sequence of τ0 -regular open sets such that x0 ∈ T∞ Since Φr (En ) = Φ(G(En )) = Φ(En ) = En for n=1 Φr (En ) for every n ∈ N. T every n ∈ N, we have that x0 ∈ ∞ n=1 En . Let {cn }n∈N ⊂ R+ be a sequence with cn & 0 and (x0 − cn , xo + cn ) ⊂ En for every n ∈ N. Putting rn = cn+1 for every S n ∈ N we have that (x0 −c1 , x0 +c1 )\{x0 } ⊂ ∞ −rn , x0 +rn ))). n=1 (En ∩(R\(x S∞ S 0 Hence x0 ∈ G( n=1 (En ∩ (R \ (x0 − rn , x0 + rn ))) = Φr ( ∞ (E n=1 n ∩ (R \ (x0 − rn , x0 + rn )))). t u Theorem 5.6. Let f : R → R, Φ ∈ Φ and x0 ∈ R. If for every decreasing seT quence {En }n∈N of τ0 -regular open sets such that x0 ∈ ∞ Φ(En ) there exists n=1 S a sequence {rn }n∈N ⊂ R+ with rn & 0 such that x0 ∈ Φ( ∞ n=1 (En ∩ (R \ [x0 − rn , x0 + rn ]))) then TΦr -topological continuity and TΦr -restrictional continuity of the function f at x0 are equivalent. Proof. It is sufficient to prove condition (b) of Theorem 5. Let {En }n∈N ⊂ Ba T be a decreasing sequence such that x0 ∈ ∞ Then {G(En )}n∈N is a n=1 Φr (En ).T decreasing sequence of regular open sets such that x0 ∈ ∞ n=1 Φ(G(En )). Hence there exists a sequence {rn }n∈N ⊂ R+ with rn & 0 such that S x0 ∈ Φ( ∞ n=1 (G(En ) ∩ (R \ [x0 − rn , x0 + rn ]))). For every n ∈ N we get G(En ∩ (R \ [x0 − rn , x0 + rn ])) = G(En ) ∩ (R \ [x0 − rn , x0 + rn ]) S∞

⊂ G(

n=1 (En ∩ (R \ [x0 − rn , x0 + rn ]))).

Hence S∞

Φ(

n=1 (G (En ) ∩ (R \ [x0 − rn , x0 + rn ])))



66

Katarzyna Flak and Jacek Hejduk

S∞

Φ (G (

n=1 (En ∩ (R \ [x0 − rn , x0 + rn ]))))

and x0 ∈ Φ(G( ∞ (E ∩ (R \ [x0 − rn , x0 + rn ])))) S n=1 n = Φr ( ∞ (E ∩ (R \ [x0 − rn , x0 + rn ]))) S n=1 n = Φr ( ∞ (E ∩ n=1 n (R \ (x0 − rn , x0 + rn )))). S

t u The converse of Theorem 5.6 is not true. Let Φ(A) = A for every A ∈ τ0 and let x0 ∈ R. Putting En = (x0 − εn , x0 + εn ), where {εn }n∈N ⊂ R+ is a sequence T tending to 0, we have x0 ∈ ∞ n=1 Φ(En ). At the same time for every sequence {rn }n∈N ⊂ R+ with rn & 0 we get that S x0 6∈ Φ( ∞ n=1 ((En ∩ (R \ [x0 − rn , x0 + rn ])))). On the other hand, by Property 2, TΦr -restrictional continuity and TΦr -topological continuity are equivalent. The following theorem establishes the equivalence in Theorem 5.6 under additional assumption. Theorem 5.7. Let Φ ∈ Φ be an operator such that Φ(A) = Φ(B) for every A, B ∈ τ0 whenever A M B is countable. Then for an arbitrary function f : R → R and x0 ∈ R, TΦr -topological continuity and TΦr -restrictional continuity of f at x0 are equivalent if and only if for every decreasing sequence {En }n∈N of τ0 T regular open sets such that x0 ∈ ∞ Φ(En ) there exists a sequence {rn }n∈N ⊂ n=1 S∞ R+ with rn & 0 such that x0 ∈ Φ( n=1 (En ∩ (R \ [x0 − rn , x0 + rn ])). Proof. Sufficiency is a consequence of the previous theorem. Necessity. Let us suppose that there exists a decreasing sequence {En }n∈N of T regular open sets such that x0 ∈ ∞ n=1 Φ(En ) and for every sequence {rn }n∈N ⊂ R+ with rn & 0, we have S x0 6∈ Φ( ∞ n=1 (En ∩ (R \ [x0 − rn , x0 + rn ]))). Let  for x 6∈ E1 and x 6= x0 ,  2 f (x) = 1/n for x ∈ En \ En+1 and x 6= x0 ,  T  0 for x ∈ ∞ n=1 En or x = x0 . Then ∀ En ⊂ {x ∈ R : | f (x) − f (x0 )| ≤ 1/n}

n∈N

and x0 ∈ Φ(En ) = Φr (En ). Thus f is TΦr -topologically continuous at x0 . Let us suppose that f is TΦr -restrictionally continuous at x0 . Then there exists a set E ∈ Ba such that x0 ∈ Φr (E) and f |E is τ0 -continuous at x0 . Hence for every n ∈ N there exists rn > 0 such that E ∩ (x0 − rn , x0 + rn ) ⊂ {x ∈ R : | f (x) − f (x0 )| ≤ 1/n}. We can assume that rn & 0. Then for every n ∈ N,

5. On equivalence of topological and restrictional continuity

67

E ∩ (R \ [x0 − rn+1 , x0 + rn+1 ]) ∩ (x0 − rn , x0 + rn ) ⊂ En+1 ∩ (R \ [x0 − rn+1 , x0 + rn+1 ]). Hence G(E) ∩ (R \ [x0 − rn+1 , x0 + rn+1 ]) ∩ (x0 − rn , x0 + rn ) ⊂ G(En+1 ) ∩ ((R \ [x0 − rn+1 , x0 + rn+1 ])). This implies that G(E) ∩

S∞

n=1 ((R \ [x0 − rn+1 , x0 + rn+1 ]) ∩ (x0 − rn , x0 + rn )) S ⊂ ∞ n=1 (En+1 ∩ (R \ [x0 − rn , x0 + rn ])) S∞ ⊂ n=1 (En ∩ (R \ [x0 − rn , x0 + rn ])).

Then S∞

Φ(G(E)) ∩ Φ(



n=1 ((R \ [x0 − rn+1 , x0 + rn+1 ]) ∩ (x0 − rn , x0 + rn )))) S Φ( ∞ n=1 (En ∩ (R \ [x0 − rn , x0 + rn ]))).

Since Φ( ∞ , x0 + rn+1 ]) ∩ (x0 − rn , x0 + rn ))) = n=1 ((R \ [x0 − rn+1 S Φ ((x0 − r1 , x0 + r1 ) \ ( ∞ n=1 {rn } ∪ {x0 })) = Φ(x0 − r1 , x0 + r1 ) ⊃ (x0 − r1 , x0 + r1 ) and x0 ∈ Φr (E) = Φ(G(E)). The contradiction that S x0 ∈ Φ ( ∞ n=1 (En ∩ (R \ [x0 − rn , x0 + rn ]))) ends the proof. S

t u

References [1] J. Hejduk, On topologies in the family of sets having the Baire property, Georgian Math. J. 22(2) (2015), 243-250. [2] J. J˛edrzejewski, On limit numbers of real functions, Fund. Math. 83 (1974), 269-281. [3] R. Johnson, E. Łazarow, W. Wilczy´nski, Topologies related to sets having the Baire property, Demonstratio Math. 22(1) (1989), 179-191. [4] J. Lukeš, J. Malý, L. Zajiˇcek, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Math. 1189, Springer–Verlag, Berlin, 1986. [5] J. C. Oxtoby, Measure and category, Springer–Verlag, Berlin, 1987. [6] W. Wilczy´nski, A category analogue of the density approximate continuity and approximate derivative, Real Analysis Exchange 10 (1984/85), 241-265. [7] W. Wilczy´nski, Density topologies, Chapter 15 in Handbook of Measure Theory, Ed. E. Pap. Elsevier, 2002, 675-702.

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[8] W. Wilczy´nski, W. Wojdowski, A category Ψ -density topology, Cent. Eur. J. Math. 9(5) (2011), 1057-1066. [9] W. Wojdowski, A category analogue of the generalization of Lebesgue density topology, Tatra Mt. Math. Publ. 42 (2009), 11-25. [10] W. Wojdowski, A generalization of the c-density topology, Tatra Mt. Math. Publ. 62 (2015), 67-87. [11] W. Wojdowski, Density topologies involving measure and category, Demonstratio Math. 22 (1989), 797-812. [12] W. Wojdowski, On a generalization of the density topology on the real line, Real Anal. Exchange 33 (2007/2008), 201-216.

K ATARZYNA F LAK Faculty of Mathematics and Computer Science, University of Łód´z Banacha 22, PL-90-238 Łód´z, Poland E-mail: [email protected]

JACEK H EJDUK Faculty of Mathematics and Computer Science, University of Łód´z Banacha 22, PL-90-238 Łód´z, Poland E-mail: [email protected]

Chapter 6

On fields inspired with the polar HSV − RGB theory of Colour

JÁN HALUŠKA

2010 Mathematics Subject Classification: 92C55, 12E30. Key words and phrases: RGB representation of colour, polar colour space, parabolic complex numbers HSV theory. Acknowledgement. This paper was supported by Grant VEGA 2/0178/14 and by the Slovak-Ukrainian joint research project "Vector valued measures and integration in polarized vector spaces".

6.1 Introduction 6.1.1 Traditional Colour theories There are more different sources of the theory of Colour which approach to the subject from different sides and are complementary in this sense. Computer graphics The HSV (Hue-Saturation-Value) theory is the most common representation of points in an RGB (Red-Green-Blue) color technical model. Computer graphics pioneers developed the HSV model in the 1970s for computer graphics applications (A. R. Smith in 1978, also in the same issue, A. Joblove and H. Greenberg). A HSV theory is used today in color pickers, in image editing software, and less commonly in image analysis and computer vision. A rather extensive explanation of the present State of Art in industry we can find in [11].

70

Ján Haluška

Biophysics Th. Young and H. Helmholtz proposed a trichromatic theory. Their theory states that the human retina contains dispersed photo-sensitive clusters, where each of these clusters consists of three types of sensitive cones which peaks in short (420–440 nm), middle (530–540 nm), and long (560– 580 nm) wavelengths. Weighting a total light power spectrum by the individual spectral sensitivities of the three types of cone cells gives three effective stimulus values; these three values make up a tristimulus specification of the objective color of the light spectrum. In Fig. 6.1, there are schematic behaviours of these sensitive clusters of cells. Fine arts For the revelatory theories of Colour written by authors who are not mathematicians (artistic photographers, visual artists), c.f. [1, 6, 7].

6.1.2 Terminology For terminology, basic and also advanced concepts about Colour, we refer to [12], Chapter 11 (Vol. III, Vision and vision optics; Chap. 11, Color vision mechanism).

6.1.3 Comments to modelling of Colour A reflected electro-magnetic vibration energy is filtered with the (human) tristimulus apparatus in the eye retina into three functions (also called the RGB stimulus curves). Here is an information loss, because we see in the wavelength interval approximately 350 nm – 750 nm although there is some power output theoretically within the whole [0, +∞). Vibrations within other vibration intervals are partially perceived by other senses (hearing, touch). Also various animals have various intervals within they can see. The vision process continues in the brain where the obtained three stimuli curves are aggregated back. The result is a registry of Colour of the object. In our theory, this aggregation is a linear combination of three basic colours (poles) with the coefficients which are complex functions defined on the interval [0, +∞) (= all possible frequencies). In Fig. 6.1, we see that realistic stimulus curves may have parts which are particularly in the negative (under the x-axis), i.e., the accession to the resulting Colour may happen also such that the parts of curves "absorbs" energy. Some practical aspects about RGB in computer graphics we can find in [10].

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Fig. 6.1 A realistically shaped tristimulus reaction on a Colour

Fig. 6.2 A horse-shoe planar cut of the visible colour space; Black = non visible

Outside a certain area of the domain, the colour aggregated from R, B, G curves is not very exact because of distorted perceptions by human senses, see Fig. 6.2. We use an imagination of the Colour Hues in the Colour wheel. The reason is that there are colour hues which are not in the linear rainbow palette (e.g., Pink). We suppose that angles of the basic RGB colours in the colour wheel

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Fig. 6.3 A Colour wheel with colours of the equal energy (the upper surface)

correspond to the angles 0 ≡ 2π, 2π/3, 4π/3, respectively, which is approximately true in reality. An incoming white sun light is Colours is decomposed into a sum of three RGB curves. So it is also in our model. But abstracting from the natural perception, the number three for basic colours is not substantial. Our polar theory works for arbitrary natural K ≥ 3. For a review, in the realm of animals, there are mono-chromatic Arctic mammals; most of mammals have sensibility only for two colours, they are dichromats; birds and insects are mostly tetrachromats. Concerning primates, the human vision is trichromatic. The record keeps the Mantis shrimp’s vision with K = 12, cf. [13]. There are also artificial colour schemes coming and used in the industry for some good reasons. E.g., we know the CMYK (cyan, magenta yellow, black), RGYB (red, green, yellow, blue) systems, and others.

6.1.4 The point and interval characteristics of light Hue is a point characteristic (it is determined in a point), the saturation and brightness are "interval" characteristics, i.e., they are determined for an interval, not at a point (similarly as the notions concavity-convexity has no sense at a points). Hue of Colour is the wavelength within the visible-light spectrum at which the energy output from a source is greatest. This is shown as the peak of the sum of the three intensity curves in the accompanying graphs of intensity versus wavelength. In the illustrative examples in the pictures Fig. 6.4, Fig. 6.5, Fig. 6.6, all 3 × 3 = nine colors there have the same Hue with a wavelength 500 nm, in the yellow-green portion of the spectrum.

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Saturation is an expression for the relative bandwidth of the visible output from a light source. In the diagram Fig. 6.4, the notion of saturation is represented relatively, by the steepness of the slopes of the curves. Here, the blue curve represents a color having the greatest saturation. As saturation increases, the color with the same Hue appears more "pure." As saturation decreases, colors appear more "washed-out." Brightness is a relative expression of the intensity of the energy output of a visible light source. It can be expressed as a total energy value (different for each of the curves in the diagram Fig. 6.5), or as the amplitude at the wavelength where the intensity is greatest. Energy is imagined as the area under the curve. In the picture the blue curve has the lowest brightness. As we can see, Saturation and Brightness are generally non-comparable parameters of Colour, cf. Fig. 6.6. One commonly supposes that all possible colours can be specified according to these three parameters and that Colors can be represented in terms of the RGB components. Thus the whole information in the RBG Colour theory is contained the three tristimulus curves. A concept of triangular coefficients mathematically reflects this idea.

6.1.5 Semi-field of triangular coefficients A semi-field X is a set equipped with an algebraic structure with binary operations of addition (+) and multiplication (·), where (X, +) is a commutative semi-group. (X, ·, 1) is a multiplicative group with the unit 1, and multiplication is distributive with respect to addition from both sides. For a review of semi-fields, c.f. [14]. A semi-field X is called a semi-field with zero if there exists an element O ∈ X such that x · O = O and x + O = x for every x ∈ X and the distributivity of multiplication from both sides is preserved for the extended system. Example 6.1. Let R+,0 := (0, ∞) ∪ {0} = [0, ∞) be a ray with all structures heredited from the real line R. This is one of trivial real semi-field with zero. Definition 6.1. We say that a set T = {q + ψ( f )ε | q ≥ 0, ψ( f ) ∈ R[0,+∞) , f ∈ [0, +∞)} is called to be the set of all triangular coefficients, where ε is the parabolic imaginary unit, |ε| = 1, ε 2 = 0, cf. [5].

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If [a + b( f )ε] ∈ T and [A + B( f )ε] ∈ T are two triangular coefficients, we define the operations of addition, multiplication, and division as following. For every a, A ≥ 0; b( f ), B( f ) ∈ R[0,+∞) , f ∈ [0, +∞), [a + b( f )ε] + [A + B( f )ε] := [a + A] + [b( f ) + B( f )]ε,

(6.1)

[a + b( f )ε] · [A + B( f )ε] := aA + [aB( f ) + Ab( f )]ε,

(6.2)

A 6= 0 =⇒

a + b( f )ε a Ab( f ) − aB( f ) ε. := + A + B( f )ε A A2

(6.3)

For the better reading, triangular coefficients are written in square brackets in the sequel of the paper. Lemma 6.1. The set T with the operations defined in (6.1), (6.2), (6.3) is a semi-field. Proof. Saving the denotation of the previous definition we see that [a + b( f )ε] + [A + B( f )ε] ∈ T , [a + b( f )ε] · [A + B( f )ε] ∈ T , a+b( f )ε and for A 6= 0, A+B( f )ε ∈ T . The assertion is an obvious enlargement of the result for parabolic-complex numbers to functions defined on the non-negative real ray. t u In our polar theory of Colour, a role of scalars will play elements of the semi-field T . Corollary 6.1. For every function B( f ), f ∈ [0, +∞), and by (6.3), A 6= 0 =⇒

1 −B( f ) 1 = + ε ∈ T. A + B( f )ε A A2

The aim of the paper To model a Colour space as a three-polar field

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450

75

550

Fig. 6.4 Monotonicity of Saturation; Brightness and Hue are fixed

300

800

Fig. 6.5 Monotonicity of Brightness; Saturation and Hue are fixed

350

650

Fig. 6.6 Non-comparable Saturation and Brightness, Hue is fixed

6.2 Mastering polar Colour spaces 6.2.1 Poles and the definition of Colour space The real line is two-polar and we are using the obvious polar operators, i.e., the signs (+) and (−). In this sense, we will also understand three signs (poles)

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R, G, B in this paper. The poles R, G, B could be understand also as the generalized signs. In fact, poles can be chosen as objects of various mathematical nature. In this paper, we use the following finite set of poles: ( √ √ ) 1 1 3 3 ı, B = − − ı , A = R = 1 + 0ı, G = − + 2 2 2 2 a set of three vertices of an equilateral triangle in the elliptic complex plane, where ı is a usual (elliptic) complex unit, ı2 = −1, |ı| = 1. Sometimes we will equivalently speak that poles are polar operators. Remark 6.1. Which properties are asked from poles in general? Various other objects of different nature can also be used as poles. E.g., functions, operators, the set A when replacing complex unit ı by the complex units ε, κ of the parabolic (ε 2 = 0) or hyperbolic (κ 2 = 1) complex units, respectively, etc. In our case, we use the operators R, G, B ∈ C which 1. are applicable to "all objects" (similarly as signs plus and minus); 2. fulfil the condition R + G + B = 0 (presence of the white point of the colour space); 3. R, G, B are different and non-collinear points; 4. a symmetry in some sense of the set A is desirable. All these terms will be precised below. Definition 6.2. Let us denote by 4 := {R[r + ρ( f )ε] + G[g + γ( f )ε] + B[b + β ( f )ε] | [r + ρ( f )ε] ∈ T, [g + γ( f )ε] ∈ T, [b + β ( f )ε] ∈ T },

(6.4)

where r, g, b ≥ 0; ρ( f ), γ( f ), β ( f ) ∈ R[0,+∞) , f ∈ [0, +∞). The set 4 is called the colour space. An element of the colour space is called to be the colour. Remark 6.2. (1) Although the functions ρ( f ), γ( f ), β ( f ) ∈ R[0,∞) are arbitrary in our paper, in the praxis they are supposed to have "good" properties, e.g., they are supposed to be unimodal, continuous, etc. (2) Elements of 4 are (mixed elliptic–parabolic) bicomplex numbers. For (elliptic–elliptic) bicomplex numbers, cf. [8].

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6.2.2 Achromatic part of Colour Every Colour is a composite of chromatic (pure colours) and achromatic (grey and noises) parts. Both parts of colour can be derived from the decomposition of the tristimulus sum into three individual curves. In this paper, the achromatic part of colour is derived from an element [a + α( f )] ∈ T , where a ∈ [0, ∞) determines a Hue of Grey.1 The function α( f ) ∈ R[0,∞) represents noise. We incorporate an achromatic part of Colour into our theory dealing with the following subsets (cuts) of the colour space 4: Definition 6.3. Let [a + α( f )ε] ∈ T and [r + ρ( f )ε] ∈ T , [g + γ( f )ε] ∈ T , [b + β ( f )ε] ∈ T . Denote by o[a+α( f )ε] := R[a + α( f )ε] + G[a + α( f )ε] + B[a + α( f ))ε], O :=

[

o[a+α( f )ε] ⊂ 4,

[a+α( f )ε]∈T

and sa := {x ∈ 4 | x = R[a + ρ( f )ε] + G[a + γ( f )ε] + B[a + β ( f )ε]}, S :=

[

sa ,

a∈[0,∞)

where a ≥ 0 and α( f ), ρ( f ), γ( f ), β ( f ) are arbitrary functions in R[0,∞) . Lemma 6.2. Let x ∈ 4. Let λ ∈ O. Then x = x+λ. Proof. From definition of O it follows that the triangular coefficients [r + ρ( f )ε], [g + γ( f )ε], [b + β ( f )ε] in x ∈ 4 are ambiguous since for every arbitrary [a + α( f )ε] ∈ T , there holds x= =

R[r + ρ( f )ε] + G[g + γ( f )ε] + B[b + β ( f )ε] R{[r + ρ( f )ε] + [a + α( f )ε]} + G{[(g + γ( f )ε] + [a + α( f )ε]} + B{[(b + β ( f )ε] + [a + α( f )ε]} = x + λ , λ ∈ O. t u

1

The so called Value, which is a term overtaken from the HSV Colour theory; not very apt for a mathematical theory.

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In the view of this lemma we introduce the following notion. Definition 6.4. Let X ∈ A. We say that colour x ∈ 4 is X-polarized if it can be expressed in the form x = X[x + ξ ( f )ε], [x + ξ ( f )ε] ∈ T . Definition 6.5. The congruence given in Lemma 6.2 is known also as the Cancellation law, c.f. [4]). So, (this holds concerning all arithmetic operations in 4 we will define in the sequel of this paper). So, we operate with the accuracy of congruent triples of triangular coefficients. Physically Cancellation law means an ambiguity with respect to the achromatic parts of Colour. This is expressed with using of the phrase "with respect to Cancellation law". For the sake of simplicity and without loss of precision, this phrase will be often omitted in the text.

6.3 Arithmetic operations in 4 Let us denote for the following sections: x = {R[r + ρ( f )ε] + G[g + γ( f )ε] + B[b + β ( f )ε]} ∈ 4, y = {R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε]} ∈ 4 be two colours.

6.3.1 Addition in 4 (Mixing of Colours) We define: x ⊕ y :=

R{[r + ρ( f )ε] + [u + σ ( f )ε]} +G{[g + γ( f )ε] + [v + χ( f )ε]} +B{[b + β ( f )ε] + [t + ξ ( f )ε]}, = R[{r + u} + {ρ( f ) + σ ( f )}ε] +G[{g + v} + {γ( f ) + χ( f )}ε] +B[{b + t} + {β ( f ) + ξ ( f )}ε].

Remark 6.3. Remind, that the result of operation of addition is with respect to Cancellation law, i.e., for every λ1 , λ2 , λ3 ∈ O, x ⊕ y = (x + λ1 ) ⊕ (y + λ2 ) = (x ⊕ y) + λ3 .

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6.3.2 Subtraction in 4 (Inverse colours) We define Subtraction in 4 as the addition of inverse elements, x y := x ⊕ ( y). The inverse elements of the basic colours are defined as follows: R[r + ρ( f )ε] := G[r + ρ( f )ε] + B[r + ρ( f )ε], G[g + γ( f )ε] := R[g + γ( f )ε] + B[g + γ( f )ε], B[b + β ( f )ε] := R[b + β ( f )ε] + G[b + β ( f )ε]. So, x :=

R{[g + γ( f )ε] + [b + β ( f )ε]} +G{[r + ρ( f )ε] + [b + β ( f )ε]} +B{[r + ρ( f )ε] + [g + γ( f )ε]}.

And the subtraction is defined then as following: x y =

{R[r + ρ( f )ε] + [g + γ( f )ε]G + B[b + β ( f )ε]} {R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε]}

:=

{R[r + ρ( f )ε] + G[g + γ( f )ε] + B[b + β ( f )ε]} ⊕{G[u + σ ( f )ε] + B[u + σ ( f )ε]} ⊕{R[v + χ( f )ε] + B[v + χ( f )ε]} ⊕{R[t + ξ ( f )ε] + G[t + ξ ( f )ε]}

=

R{[r + ρ( f )ε] + [v + χ( f )ε] + [t + ξ ( f )ε]} +G{[g + γ( f )ε] + [u + σ ( f )ε] + [t + ξ ( f )ε]} +B{[b + β ( f )ε] + [u + σ ( f )ε] + [v + χ( f )ε]}.

Remark 6.4. We can replace polar operators R, G, B with their inverse operators √ √ C := R = −1, M := G = 1/2 − ( 3/2))ı,Y := B = 1/2 + ( 3/2))ı. This way we obtain the C, M,Y (cyan - magenta - yellow) colour scheme. These colour systems are mathematically equivalent, but to White should correspond Black as the inverse Colour. But Black does not physically exist in the electro-magnetic spectrum as Colour (in the RGB scheme, Black means an absence of energy). Therefore in praxis (e.g. in the print industry), Black

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is artificially added to the CMY system to have CMY K system (the character K is added as the abbreviation for Black). Subsuming the previous two sections, the following lemma holds: Lemma 6.3. The triple (4, ⊕, O) is an Abel additive group with respect to Cancellation law.

6.4 Multiplication in 4 New transformations of Colours For simple mixing of two Colours, it is enough to deal with the additive group of Colours. However, there are theoretical transformations of Colours which can be called as multiplications according their mathematical properties. The author did not know about any appearance of these operations in the praxis. However, using a computer digitalization, this theory enables, we can artificially produce and explore these Colours. This section is about multiplication of Colours in the Colour space 4 and about division in the factorized Colour space 4|S where S is the ideal of singular elements for division of Colours, cf. Section 6.4.3.

6.4.1 Cyclic compositions of polar operators in C It is easy to see that for the number 3 (R, G, B), there are possible six symmetric Latin squares which respectively yield 6 commutative operations ("multiplications") ⊗i : A × A → A, i = 1, 2, . . . , 6 of Colours. ⊗1 R G B

RGB RGB , GBR BRG

⊗2 R G B

RGB GBR , BRG RGB

⊗3 R G B

RGB BRG , RGB GBR

⊗4 R G B

RGB RBG , BGR GRB

⊗5 R G B

RGB BGR , GRB RBG

⊗6 R G B

RGB GRB . RBG BGR

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In this paper, we will deal only with commutative operations ⊗ which have the property ∃Y ∈ A ∀X ∈ A | Y ⊗ X = X. Such are cases ⊗1 , ⊗2 , ⊗3 . For the reason of cyclic change, we will only deal with the Latin square ⊗ = ⊗1 . So, let us define the compositions of poles ⊗ : A⊗A → A with the following Latin square table. ⊗RGB R RGB GGBR B BRG The operation of multiplication in 4 is defined in accord both with the table of the composition ⊗ and the multiplication in the semi field T . Namely, ! ! x y =



A(i) [xi + ξi ( f )ε]

A( j) [y j + χ j ( f )ε]

j=1,2,3

i=1,2,3

:=



∑ ∑

h i A(i) ⊗ A( j) [xi + ξi ( f )ε] · [y j + χ j ( f )ε]

∑ ∑

h i A(i) ⊗ A( j) [xi y j + {xi χ j ( f ) + y j ξi ( f )}ε],

i=1,2,3 j=1,2,3

=

(6.5)

i=1,2,3 j=1,2,3

where A(i) , A( j) ∈ A, i, j = 1, 2, 3; x = R[xi + ξi ( f )ε] + G[x2 + ξ2 ( f )ε] + B[x3 + ξ3 ( f )ε] ∈ 4; y = R[y1 + χ1 ( f )ε] + G[y2 + χ2 ( f )ε] + G[y3 + χ3 ( f )ε] ∈ 4. Note that multiplication in Equation (6.5) is parabolic complex. The proof of the following lemma is evident. Lemma 6.4. 1. The result of operation of multiplication is with respect to Cancellation law, i.e., x y = (x ⊕ λ1 ) (y ⊕ λ2 ) ⊕ λ3 , for every λ1 , λ2 , λ3 ∈ O. 2. The element 1 := R[1 + 0( f )ε] + G[0 + 0( f )ε] + B[0 + 0( f )ε] ∈ 4 is an unit element for the operation of multiplication in 4 (and hence also 1 + λ , λ ∈ O, with respect to the congruence given with Cancellation law).

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6.4.2 Conjugation in 4 (Polarization of the light) To define an operation of division, we introduce an operation of conjugation. Remark 6.5. Conjugation physically means a polarization of light; according to the chosen Latin square ⊗1 , a projection will be done to the R-axis. Definition 6.6. Let x = R[r + ρ( f )ε] + G[g + γ( f )ε] + B[b + β ( f )ε] ∈ 4. We say that an element x ∈ 4 is a conjugation of the element x if x := R[r + ρ( f )ε] + G[b + β ( f )ε] + B[g + γ( f )ε] ∈ 4. Theorem 6.1. Let x ∈ 4 be as in previous definition, let y = [R(u + σ ( f )ε) + G(v + χ( f )ε)] + B(t + ξ ( f )ε) ∈ 4. Then 1. x = x ∈ 4, 2. x ⊕ y = x ⊕ y ∈ 4, 3. x y = x y ∈ 4, 4. y y = RΘ , where h 2 2 +(t−u)2 Θ = (u−v) +(v−t) 2 +{(u − v)(σ ( f ) − ξ ( f )) i +(v − t)(ξ ( f ) − χ( f )) + (t − u)(χ( f ) − ξ ( f ))}ε ∈ T. Proof. The proofs of items 1., 2., 3. are exercises in algebra, we let them to the reader. We prove the last statement 4. We have: y y =

{R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε]} {R[u + σ ( f )ε] + G[t + ξ ( f )ε] + B[v + χ( f )ε]} =

using the composition table of poles and the distributive law, we continue:  = R [u + σ ( f )ε]2 + [v + χ( f )ε]2 + [t + ξ ( f )ε]2  +G [u+σ ( f )ε][v+χ( f )ε]+[u+σ ( f )ε][t+ξ ( f )ε]+[v+χ( f )ε][t+ξ ( f )ε]  +B [u+σ ( f )ε][v+χ( f )ε]+[u+σ ( f )ε][t+ξ ( f )ε]+[v+χ( f )ε][t+ξ ( f )ε] = By the definition of subtraction, h = R {[u + σ ( f )ε]2 + [v + χ( f )ε]2 + [t + ξ ( f )ε]2 }

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−{[u + σ ( f )ε][v + χ( f )ε] +[u + σ ( f )ε][t + ξ ( f )ε] i +[v + χ( f )ε][t + ξ ( f )ε]} = Now, we have to show that this is a R-polarized element. Indeed, we continue: n = R [u + σ ( f )ε]2 + [v + χ( f )ε]2 + [t + ξ ( f )ε]2 −[u + σ ( f )ε][v + χ( f )ε] −[v + χ( f )ε][t + ξ ( f )ε] o

(6.6)

−[t + ξ ( f )ε][u + σ ( f )ε] =  n1 1 2 2 =R [u + σ ( f )ε] − [u + σ ( f )ε][v + χ( f )ε] + [v + χ( f )ε] 2 2   1 1 2 2 + [u + σ ( f )ε] − [u + σ ( f )ε][t + ξ ( f )ε] + [t + ξ ( f )ε] 2 2  o 1 1 2 2 + [v + χ( f )ε] − [v + χ( f )ε][t + ξ ( f )ε] + [t + ξ ( f )ε] 2 2 h 2 = R2 { [u + σ ( f )ε] − [v + χ( f )ε]  2 (6.7) + [u + σ ( f )ε] − [t + ξ ( f )ε] o  2 + [v + χ( f )ε] − [t + ξ ( f )ε] n = R2 [{u − v} + {σ ( f ) − χ( f )}ε]2 + [{v − t} + {χ( f ) − ξ ( f )}ε]2 o (6.8) +[{t − u} + {ξ ( f ) − σ ( f ))}ε]2 h 2 2 +(t−u)2 = R (u−v) +(v−t) 2 +{(u − v)(σ ( f ) − ξ ( f ))

(6.9)

i +(v − t)(ξ ( f ) − χ( f )) + (t − u)(χ( f ) − ξ ( f ))}ε , Since (u − v)2 + (v − t)2 + (t − u)2 ≥ 0, then y y = RΘ , where Θ ∈ T .

t u

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6.4.3 The ideal S We proved that y y is an R-polarized element in 4. But what about elements y ∈ 4 such that y y ∈ O and y 6∈ O? Definition 6.7. An element y ∈ 4 such that y y = O and y 6= O is called a singular element. The set of all singular elements (including elements of O) is denoted by H. Lemma 6.5. H = S. Proof. Let y = R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε] ∈ 4. From Equation (6.9) it follows that y y ∈ O if and only if u = v = t for arbitrary real functions σ ( f ), χ( f ), ξ ( f ) ∈ R[0,+∞) , and every u = v = t ≥ 0. t u Corollary 6.2. Since the result t = u = v of the previous Lemma proof is symmetrical and the analogical result can be obtained using with the cyclic change for the Latin squares (⊗1 ), ⊗2 , ⊗3 , the ideal S of all singular elements for divisions derived from matrices ⊗1 , ⊗2 , ⊗3 is the same. Theorem 6.2. S is a two-sided ideal in the ring (4, ⊕, , O) with respect to operation of multiplication with respect to Cancellation law. Proof. We have to proof: O & S & 4, x ∈ S & y ∈ S =⇒ x + y ∈ S, and x ∈ S & y ∈ 4 =⇒ x y ∈ S. The first two assertions are evident. Prove the third assertion. Let x = R[a + ρ( f )ε] + G[a + γ( f )ε] + B[a + β ( f )ε] ∈ S and let y = R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε] ∈ 4. We have: x y = {R[a + ρ( f )ε] + G[a + γ( f )ε] + B[a + β ( f )ε]} {R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε]} = since x ∈ S, = {Rρ( f )ε + Gγ( f )ε + Bβ ( f )ε} {R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε]} =

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since ε 2 = 0, = ε{Rρ( f ) + Gγ( f ) + Bβ ( f )} {Ru + Gv + Bt} = using the composition table of poles, = ε{R[uρ( f ) + vβ ( f ) + tγ( f )] + G[vρ( f ) + uγ( f ) + tβ ( f )] + B[tρ( f ) + vγ( f ) + uβ ( f )]}. t u

We obtained an element in S.

6.4.4 Division in the factor space 4|S Let x, y ∈ 4 and y ∈ / S. Remark 6.6. The unit element 1 = R[1 +0( f )ε]+G[0 +0( f )ε]+B[0 +0( f )ε] is a mathematical abstraction. In the real world, there is no tristimulus of this kind. However, using this theoretical object 1, we are able to divide real Colours. Division is defined as follows: x y := (x 1) y = x (1 y). Find the element 1 y ∈ 4 if it exists. Theorem 6.3. Let y = R[u + σ ( f )ε] + G[v + χ( f )ε] + B[t + ξ ( f )ε] 6∈ S. Then 1 y = {R[u + σ ( f )ε] + G[t + ξ ( f )ε] + B[v + χ( f )ε]} · (2/Θ ), where Θ is defined in (6.9). Proof. By Theorem 6.1, the item 4., and Lemma 6.5, 1 y =

y . y y t u

From Theorem 6.3 it follows that we can divide every colour by another colour but elements in the ideal S.

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6.4.5 Compatibility of the additive and multiplicative structures of 4 To be complete in verifying of the axioms of the field, we bring the following more-less evident lemma. Lemma 6.6. 1. 1 6∈ S; 2. 1 1 = 1; 3. Let λ ∈ O. Then 1 6= λ ; 4. Let x, y, z ∈ 4. Then x (y ⊕ z) = (x y) ⊕ (y z). Proof. (1) (2) The first and second items are trivial. (3) Let [a + α( f )ε] ∈ T . Then 1 = R[(1 + a) + α( f )ε] + G[a + α( f )ε] + B[a + α( f )ε] 6= R[a + α( f )ε] + G[a + α( f )ε] + G[a + α( f )ε] ∈ O. (4) The fourth item (distributive law) is ensured by construction of the additive ⊕ and multiplicative operations in the set 4. The verification of this equation is elementary and we let it to the reader as an exercise. t u

6.5 Mathematical subsuming We collect mathematical results of the paper into the following theorem. Theorem 6.4. Let A ⊂ C be a set of three poles. Let T be a semi-field of triangular coefficients, cf. Subsection 6.1.5. For a fixed Latin square ⊗ ∈ {⊗1 , ⊗2 , ⊗3 }, cf. Section 6.4.1, the system (4, ⊕, , O, 1) (called the Colour space) is an commutative Abel ring (with respect to Cancellation law congruence) and particular division, c.f. Subsection 6.2.1. There are two Abel groups: an additive group (4, ⊕, O) (with respect to Cancellation law). The second group (4|S, , 1) (with respect to Cancellation law) is a multiplicative group with the unit 1, where the set S, see Lemma 6.5, is an ideal, cf. Section 6.2.2.

6. On fields inspired with the polar HSV − RGB theory of Colour

87

The additive and multiplicative groups are linked together by the distributive law of addition with respect to the multiplication/division commutatively from both sides. The structure (4|S, ⊕, , S, 1) is an field (with respect to Cancellation law).

6.6 Conclusions We created and described a mathematical theory of Colour space which factorized with the ideal S is a tripolar RGB field with respect to Cancellation law. This theory has practical and theoretical application to everything where the phenomenon Colour is sofisticated on the RGB language. For practical applications, the model needs to include a more thorough theory of the achromatic part of Colour and also it supposed to take into account some corrections implied from the technical equipment limitations and human sensory distortions.

References [1] D. Briggs, The Dimensions of Color, e-book, Art gallery of New South Wales; Julian Ashton Art Gallery - Sydney; National Art School - Sydney; copyrighted for years 2007-2013, www.huevaluechroma.com [2] J. S. Golan, Some recent applications of semiring theory, Int. Conf. on Algebra in memory of Kostia Beidar, National Cheng Kung University, Tainan, March 6-12, 2005, pp. 18. [3] T. Gregor, Three-polar space over the semi-field of double numbers, Tatra Mount. Math. Publ. 61 (2014), 167-173. [4] T. Gregor, J. Haluška, Lexicographical ordering and field operations in the complex plane, Stud. Mat. 41 (2014), 123-133. [5] A. A. Harkin, J. B. Harkin, Geometry of general complex numbers, Mathematics magazine, 77 (2004), 118–129. [6] R. Hirsch, Exploring Colour Photography: A Complete Guide. Laurence King Publishing, 2004. ISBN 1-85669-420-8. [7] R. W. G. Hunt, The Reproduction of Colour (6th ed.). Chichester UK: Wiley, IS & T Series in Imaging Science and Technology. 2004. [8] M. E. Luna-Elizarrarás, M. Shapiro, D. C. Struppa, A. Vajiac Schmid, Bicomplex Numbers and their Elementary Functions. CUBO A Mathematical Journal 02(2012), 61-80. [9] E. Ružický, A. Ferko, Computer graphics and image processing (in Slovak). Sapientia, Bratislava 1995, ISBN 80-967180-2-9, 325 pp. + ix. [10] D. Pascale, A review of RGB color spaces ... from xyY to R0 G0 B0 , The Babel Color Co., Montreal 2002. (revised 2003).

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[11] R. M. Soneira, Display technology shoot-out: comparing CRT, LCD, plasma and DLP displays, 1990-2005, Parts: Overview, I., II., III., IV. http://www.displaymate.com (Part II: Gray-Scale and Color Accuracy); copyrighted for years 1990-2005. [12] A. Stockman, D. H. Brainard, Color vision mechanisms, In M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. van Stryland (Eds.), The Optical Society of America Handbook of Optics, 3rd edition, Volume III: Vision and Vision Optics., McGraw Hill, New York 2010. [13] H. H. Thoen, M. J. How, T. H. Chiou, J. Marshall, A different form of color vision in mantis shrimp, J. Science 343 (2014), 411-413. [14] E. M. Vechtomov, A. V. Cheraneva, Semifields and their properties, Jour. of Math. Sciences, Vol. 163 no. 6, 2009; translated from Fundamental’naya i prikladnaya matematika, Vol. 14 no. 5, 3-54, 2008 (in Russian). [15] A learning community for photographers Cambridge Colours, e-tutorial on Colour c 2015, www.cambridgeincolour.com photography,

JÁN H ALUŠKA Mathematical Institute, Slovak Academy of Sciences Grešákova 6, 040 00 Košice, Slovakia E-mail: [email protected]

Chapter 7

Generalized (topological) metric space. From nowhere density to infinite games

EWA KORCZAK-KUBIAK, ANNA LORANTY AND RYSZARD J. PAWLAK

2010 Mathematics Subject Classification: 54A05, 91A05, 91A06, 54C60. Key words and phrases: generalized topological space, Baire property, generalized metric space, set valued function, (strongly) transitive set valued function, Banach-Mazur game, set function game.

7.1 Introduction and basic notations and denotations In many considerations connected with pure mathematics as well as with its applications, topological structures play an important role. Without them, applying of many mathematical tools would be impossible. For that reason, having defined some set (sometimes in very practical situations connected for example with information flow theory, graph theory etc.) we tend to equip this set with some topological structure (e.g. topology, metric, pseudometric, uniformity etc.). It is also a natural action to enrich possibility of creating such structures. At the end of XX century Á. Császár introduced new structures, generalized topologies ([9, 10]). In the paper [21] the possibility of applying these structures in research connected with information flow has been noticed for the first time.

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From the point of view of pure mathematics, considering generalized topological spaces gives new research possibilities, which often do not have its analogues in the case of classical topological spaces. The authors of this chapter have focused their examinations on the notion of nowhere density of sets. The reason for that is the fact that adopting the two definitions which give equivalent notions in classical topological spaces, leads to nonequivalent notions in the case of generalized topological spaces. In consequence, examining of different notions connected with analogues of meager sets and further with Baire spaces, is desirable. Analysis of properties of these sets or spaces gives completely new possibilities and allows to obtain new theorems which are unknown in classical case. The authors of this chapter have devoted to this issue, among others, the papers: [24, 22]. Let us return for a moment to classical topological spaces. The notion of a Baire space is closely related to the Baire Theorem for complete spaces. From this fact, the new challenge arises: building theory of generalized metric spaces and considering in it the issue of nowhere density of sets and, in consequence, complete spaces, Baire spaces and analogues of known theorems for complete metric spaces, especially connected with the Cantor Theorem and infinite games. Initial results within the scope of this issue were published in [19]. Significant development of this theory is submitted in, unpublished yet, paper [25]. The above introduction justifies the fact that we will start our considerations here with presenting basic facts connected with nowhere density in generalized topological space and Baire generalized topological space. It is not our aim to extend excessively all the research directions connected with this issue but only to signalize basic definitions and theorems. By contrast, we will discuss more precisely the issue related to generalized metric spaces. Particular attention will be paid to infinite games. We will start with the generalization of known Banach-Mazur Game and then we will show a new original game which has been presented for the first time in [19]. In order to avoid excessive lengthening of this section we will not present the proofs of theorems as well as examples. All the facts presented here one can find in [19, 24, 22]. Throughout the paper N denotes the set of positive integers. The symbol N0 stands for the set N ∪ {0}. We will write ρE for the Euclidean metric for real line. The power set of a nonempty set X will be denoted by P(X). Moreover, we will denote by Γ ( f ) the graph of a function f : X → X. The symbol Θ f (x0 ) stands for the orbit of f at x0 i. e. Θ f (x0 ) = {x0 , f (x0 ), f 2 (x0 ), . . . }.

7. Generalized (topological) metric space. From nowhere density. . .

Let {Fn }n∈N ⊂ P(X). If

∞ T ∞ S

Fn+k =

n=1 k=1 ∞ S ∞ T

∞ S ∞ T

91

Fn+k , then we will say that

n=1 k=1

Fn+k is a limit of the sequence {Fn }n∈N (denoted by Lim Fn ). n→∞

n=1 k=1

Let (X, ρ) be a metric space. We will use the symbols diamρ (A), intρ (A) and clρ (A) to denote the diameter, the interior and the closure of the set A ⊂ X, respectively. Moreover, we will write ρ − lim xn = x if the sequence {xn }n∈N ⊂ n→∞ X converges to x ∈ X with respect to the metric ρ. In our consideration, set valued functions (known also under the name multifunctions) will play an important role. From now on, we will consider only set valued functions F : X ( X such that F(x) 6= 0/ for each x ∈ X. S F(a). Moreover, we put F0 (x) = If A ⊂ X and F : X ( X, we set F(A) = a∈A

{x} and Fi (x) = F(Fi−1 (x)) for i ∈ N. The notation F @ F1 (where F, F1 : X ( X) means that F(x) ⊂ F1 (x) for any x ∈ X. We will say that a sequence of set valued functions {Fn }n∈N is decreasing if Fn+1 @ Fn for n ∈ N. Let F : X ( X and Fn : X ( X(n ∈ N) be set valued functions. A sequence {Fn }n∈N is said to be s-convergent to a set valued function F (denoted F = LIM Fn ), if F(x) = Lim Fn (x) for any x ∈ X. n→∞

n→∞

7.2 GTS and GMS As it has been already mentioned, the notion of a generalized topological space was introduced by A. Császár in [9]. Generalized metric spaces were first considered in [19]. In this section we will recall basic definitions and facts connected with these notions.

7.2.1 Generalized topological space Let X be a nonempty set. We shall say that a family G ⊂ P(X) is a generS alized topology in X iff 0/ ∈ G and Gt ∈ G whenever {Gt : t ∈ T } ⊂ G . In t∈T

further considerations we will assume that G contains at least one nonempty set. The pair (X, G ) will be called a generalized topological space (briefly GTS). Moreover, if X ∈ G we shall say that (X, G ) is a strong generalized topological space (sGTS for short)) and G is a strong generalized topology.

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Let us say that B ⊂ G is a base for G if every A ∈ G is a union of elements of B ([13]). If (X, GX ) and (Y, GY ) are GTS, then a product generalized topology GX×Y in X ×Y is a collection of all sets being a union of sets of the form M1 × M2 where M1 ∈ GX and M2 ∈ GY ([15]). From now on, if we will consider a generalized topological space (X, G ), then we will use the symbol Ge to denote the family G \ {0}. / Generalized topological spaces were studied by many mathematicians (e.g. [10] - [15], [5, 19, 21, 27]). These studies are associated with pure mathematics, as well as with the applications, e.g. in the theory of information flow. It seems interesting to note that every generalized topology in X can be associated with a monotonic map Ψ : P(X) → P(X) (i.e a map such that Ψ (A) ⊂ Ψ (B) if A ⊂ B ⊂ X). More precisely, in [10] one can find that every generalized topology G in X can be generated by some monotonic map Ψ : P(X) → P(X) in the following way G = {A ⊂ X : A ⊂ Ψ (A)}. On the other hand, if Ψ : P(X) → P(X) is a monotonic map then GΨ = {A ⊂ X : A ⊂ Ψ (A)} is a generalized topology ([9]). In the theory of a generalized topological space almost all notions are defined similarly as for a standard topological space. We recall some of them since they will be useful in the next part of this note. We shall follow the terminology of [9, 10, 22]. Let (X, G ) be a generalized topological space. The G -closure (G -interior) of A ⊂ X will be denoted by cl(A) (int(A)). A set A ⊂ X is called dense if cl(A) = X. It is easily seen that A is a dense set iff for any U ∈ Ge we have that A ∩U 6= 0. / The space (X, G ) is said to be thick if for any U ∈ Ge and any finite set A ⊂ U there exists V ∈ Ge such that V ⊂ U \ A. However, despite identical definitions, the properties of some mathematical objects in the case of usual topological space may be quite different from the properties of respective objects in generalized topology. The examples of such situation are the notions of nowhere dense sets. Let (X, G ) be GTS. If int(cl(A)) = 0/ then we shall say that A is a nowhere dense set. In the case of topological space the above definition is equivalent to the fact that every nonempty open set U contains nonempty open subset V such that V ∩A 6= 0. / A simple example (see [19]) leads us to the conclusion that in the case of GTS this equivalence is false. Consequently, we have a second notion connected with nowhere density. We shall say that A ⊂ X is a strongly nowhere dense set if for any U ∈ Ge there exists V ∈ Ge such that V ⊂ U and V ∩ A = 0. /

7. Generalized (topological) metric space. From nowhere density. . .

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Of course, if A is a strongly nowhere dense set then A is a nowhere dense set. Let us note the basic difference between these concepts. Proposition 7.1 (Property 2.4 [19]). There exists GTS (X, G ) and nowhere dense sets A, B ⊂ X such that A ∪ B is not a nowhere dense set but for every two strongly nowhere dense sets A and B in an arbitrary GTS (X, G ) the union A ∪ B is a strongly nowhere dense set. An obvious consequence of the foregoing is the fact that there are two types of definitions corresponding to a meager set in usual topology. We shall say that A ⊂ X is a meager (s-meager) set if there exists a sequence {An }n∈N of S An . A set A is nowhere dense (strongly nowhere dense) sets such that A = n∈N

called a second category (s-second category) set if it is not a meager (s-meager) set. A set A is said to be residual (s-residual) if X \ A is meager (s-meager). A further consequence is a distinction of three types of notions corresponding to a Baire space in the case of topological spaces. We will say that GTS (X, G ) is • a weak Baire space if each set U ∈ Ge is an s-second category set; • a Baire space if each U ∈ Ge is a second category set; • a strong Baire space if V1 ∩ · · · ∩ Vn is a second category set for any V1 ,V2 , . . . ,Vn ∈ G such that V1 ∩ · · · ∩Vn 6= 0. / We have Theorem 7.1 (Property 2.7 [19]). If GTS (X, G ) is a strong Baire space, then it is a Baire space. If GTS (X, G ) is a Baire space, then it is a weak Baire space. The converse implications do not hold. The definition of a strong Baire space inspires us to consider a new property of GTS (X, G ): (INT-GTS) int(V1 ∩ V2 ∩ · · · ∩ Vm ) 6= 0/ for any m ∈ N and V1 ,V2 , . . . ,Vm ∈ G such that V1 ∩V2 ∩ · · · ∩Vm 6= 0. / Taking into account the above condition we have two dual theorems. Theorem 7.2 (Property 2.2 [19]). If GTS (X, G ) satisfies the condition (INTGTS) then a set A is nowhere dense if and only if it is strongly nowhere dense. Theorem 7.3 (Property 2.8 [19]). If GTS (X, G ) satisfies the condition (INTGTS) then three notions: a strong Baire space, a Baire space and a weak Baire space are equivalent.

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Certainly, since each topological space satisfies the condition (INT-GTS), then it follows that in the case of topological space the above three notions are equivalent. In this paper we will concentrate mostly on generalized metric spaces and their properties. Therefore we will only signal some facts connected with Baire generalized topological spaces. Let us start with recalling some notions. Let (X, G ) be GTS. If f : X → X, then the set of all its continuity points will be denoted by C( f ). We shall say that f is a cm-function if the set X \ C( f ) is a countable set and f −m (x) = {z ∈ X : f m (z) = x} is a meager set for any x 6∈ C( f ) and m ∈ N0 . Let F : X ( X be a set valued function. We will say that F is lower semicontinuous at a point x ∈ X if for every set U ∈ G such that F(x) ∩U 6= 0/ there exists V ∈ G such that x ∈ V and F(t) ∩ U 6= 0/ for any t ∈ V . A set valued function F is lower semicontinuous if it is lower semicontinuous at each point x ∈ X. An orbit of x0 under F is a set (sequence1 ) ΘF (x0 ) = {x0 , x1 , x2 , . . . } such that xi ∈ F(xi−1 ) for any i = 1, 2, 3, . . . . Clearly, there may exist a lot of different orbits of x0 under F. Let us denote by ΘFa (x0 ) the family of all orbits ΘF (x0 ) of x0 under F. A set valued function F is transitive if, for any pair U,V ∈ Ge, there exists a positive integer n such that V ∩ Fn (U) 6= 0. / As it happens often in the case of GTS one can consider also a dual notion. A set valued function F is strongly transitive if, for any pair U,V ∈ Ge, the set {x ∈ U : ∃ ΘΦ (x) ∩V 6= 0} / is of the second category. ΘF (x)

Then we have two interesting theorems. Theorem 7.4 (Theorem 5 [22]). Let (X, G ) be a Baire generalized topological space with a countable base. Let F : X ( X be a lower semicontinuous set valued function. The following conditions are equivalent (i) F is transitive, S (ii) the set {x ∈ X : cl( ΘFa (x)) = X} is residual. Theorem 7.5 (Theorem 10 [22]). Let (X, G ) be a thick, strong Baire generalized topological space with countable base. Let f : X → X be a cm-function and f¯ : X ( X be a set valued function such that f¯(x) = {α ∈ X : (x, α) ∈ cl(Γ ( f ))} for any x ∈ X. The following conditions are equivalent: (a) f is strongly transitive, 1

In the literature the notion of orbit is used interchangeably in both senses: as a set and as a sequence.

7. Generalized (topological) metric space. From nowhere density. . .

95

(b) there exists x0 ∈ X such that Θ f (x0 ) is a dense set and Θ f (x0 ) ⊂ C( f ), (c) f¯ is strongly transitive, (d) there exists x0 ∈ X such that there exists an orbit Θ f¯(x0 ) which is a dense set and Θ f¯(x0 ) ⊂ C( f ). More information about Baire GTS one can find in [22, 19] and [25].

7.2.2 Generalized metric space In the case of topological spaces, a special role is played by metrizable spaces and, therefore, by metric spaces. A possibility of considering abstract distances between elements of a space, allows to make a detailed analysis of many important problems in theoretical context as well as in practical issues. On the other hand, use of techniques which are specific to metric spaces, makes many considerations more simple (a proper example here is completeness of a space) or easier in description. In 2013, there was published the paper [19] in which the notion of a generalized metric space being an analogue to metric spaces for GTS was introduced. Expanding of these issues can be found in [25]. Now, we will present briefly some facts connected with this theory. Let X 6= 0. / The symbol π stands for the family of metrics defined on subsets of X, i.e. if ρ ∈ π it means that one can find a nonempty set Aρ ⊂ X such that ρ is a metric on Aρ . The set Aρ is named a domain of ρ. We will use the symbol dom(ρ)) to denote the domain of a metric ρ. The space (X, π) is called a generalized metric space (GMS for short). If we will write πX it means that for each metric ρ ∈ πX we have that dom(ρ) = X. We will say that a set A ⊂ X is π-open if for each x ∈ A there exist ρ ∈ π and ε > 0 such that x ∈ dom(ρ) and the set Bρ (x, ε) = {y ∈ dom(ρ) : ρ(x, y) < ε} is contained in A. We will denote by Gπ the family of all π-open sets in (X, π). It is easy to check that if (X, π) is GMS then (X, Gπ ) is GTS. For our further considerations, the notion of kernel of GMS will be particularly important. Let (X, π) be GMS. A kernel of the space (X, π) is a finite family π0 ⊂ π such that for any set V ∈ Geπ there exist ρ ∈ π0 with property intρ (V ) 6= 0. / If a finite family π0 ⊂ π have the property: for any V1 , ...,Vm ∈ Gπ such that V1 ∩ ... ∩ Vm 6= 0/ there exists ρ ∈ π0 such that intρ (V1 ∩ ... ∩ Vm ) 6= 0, / then we call it a perfect kernel of the space (X, π). The set of all kernels (perfect kernels) of the space (X, π) will be denoted by Ker(X, π) (Ker p (X, π)).

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Obviously, each perfect kernel of the space (X, π) is a kernel of this space. Moreover, if π0 is a perfect kernel of the space (X, π) and π1 is a finite family such that π0 ⊂ π1 ⊂ π then π1 is a perfect kernel of the space (X, π). The examples of kernels of generalized metric spaces one can find in [19] and [25]. The definitions introduced for GTS may be adopted for GMS. So we have: Theorem 7.6 (Lemma 4.4 [19]). If GMS (X, π) has a perfect kernel then it fulfills the condition (INT-GTS). We have also Theorem 7.7 (Proposition 4.5 [19]). Let (X, π) be GMS with a perfect kernel. The following conditions are equivalent: (i) (X, π) is a strong Baire space. (ii) (X, π) is a Baire space. (iii) (X, π) is a weak Baire space.

7.2.3 Complete spaces In the theory of metric spaces, a particular role is played by complete spaces. The Baire Theorem (vide definition of a Baire space), the Cantor Theorem and the Banach Fixed-Point Theorems and Banach spaces are classical examples of a wide usage of these spaces in considerations of various problems. Within the range of GTS theory one can also consider complete spaces, wherein in this case we need to consider several types of these spaces (similarly to Baire spaces). At the current stage of research, many questions related to these spaces have not been answered yet. Therefore, we will only signal the existing results concerning these issues. We shall say that GMS (X, π) is weakly complete (complete) if there exists π0 ∈ Ker(X, π) (π0 ∈ Ker p (X, π)) consisting of complete metrics. Moreover, if (X, π) is a complete space and π is a finite family consisting of complete metrics then we shall say that (X, π) is strongly complete. Obviously if (X, π) is a strongly complete space then it is a complete space, and if it is a complete space then it is a weakly complete space. Moreover, these implications can not be inverted. One can find relevant examples in [19]. The basic question one can ask, concerns the possibility of transfering the Baire Theorem for the case of GMS. In our situation, it refers to establishing

7. Generalized (topological) metric space. From nowhere density. . .

97

the relation between complete spaces and Baire spaces in GMS. This relation is established by the following theorem. Theorem 7.8 (Theorem 4.11 and Corollary 4.12 [19]). (i) If GMS (X, πX ) is weakly complete then (X, GπX ) is a weak Baire GTS. (ii) If GMS (X, πX ) is complete then (X, GπX ) is a strong Baire GTS. Another interesing question concerns analogue of the Cantor Theorem. In this case we have: Theorem 7.9 (Theorem 4.9 [19]). Let (X, π) be GMS. The space (X, π) is weakly complete if and only if there exists π0 ∈ Ker(X, π) such that for any sequence of metrics {ρn }n∈N ⊂ π0 and for any decreasing sequence of sets {Fn }n∈N such that Fn = clρn (Fn ) for n ∈ N and ρE − lim diamρn (Fn ) = 0 we n→∞

have that

∞ T

Fn is a singleton.

n=1

7.3 Infinite games The history of infinite games is quite rich. Undoubtedly, its background are finite games being considered in XVII century. The basis of strategic and positional games have been developed by Borel [2, 3, 4], von Neumann [23] and Steinhaus [28]. We do not tend to present all aspects of this issue (it would be impossible in view of a very rich literature), however it should be emphasized that it is still examined by many scientists. As examples we can mention here the following papers: [6, 7, 8, 16, 17, 18, 20, 31, 32], however mentioning these few items definitely does not exhaust the subject. In our case, we will refer the considerations connected with infinite games exclusively to GMS (X, πX ) having perfect kernel π0 .

7.3.1 A B-M game In the period 1935-1941 in the town Lwów (which at that time was in Poland), so-called Scottish Book ([30]) was created. A group of mathematicians (among others St. Banach, H. Steinhaus, S. Mazur, S. Ulam) used to meet and discuss on mathematics in the Scotish Caffé. They had written down mathematical problems in a thick notebook which was a gift from the wife of Stefan Banach.

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After some time, the notebook was named a Scotish Book (after the name of cafe). Problem 43 in that book (formulated by S. Mazur) was connected with (using modern terminology) Banach-Mazur game (we will denote it briefly BM game). An interesting history of research dealing with Banach-Mazur game can be found in [26] and [29]. Now, we will present B-M game for GMS. Let us fix GMS (X, πX ) which has a perfect kernel. Put Γ0 = X. Two players take part in the game. Let us denote them by A and B (similarly as S. Mazur had done it). The players choose sets succesively according to the following rules:

The first player chooses a set Γ1 ∈ Geπ such that Γ1 ⊂ Γ0 . Γ1

The second player chooses a set Γ2 ∈ Geπ such that Γ2 ⊂ Γ1 .

The first player chooses a set Γ3 ∈ Geπ such that Γ3 ⊂ Γ2 . Γ1

The second player chooses a set Γ4 ∈ Geπ such that Γ4 ⊂ Γ3 , etc.

Obviously {Γn }n∈N0 is a decreasing sequence of nonempty Gπ -open sets. It is easy to see that ∞ \ n=0

Γn =

∞ \

Γ2n =

n=0

There are two possibilities either

∞ \

Γ2n−1 .

n=1 ∞ T

Γn 6= 0/ and then the player A wins or

n=0 ∞ T

Γn = 0/ and in this case, the player B is a winner.

n=0

In order to establish a definition of a strategy and a winning strategy, we will formulate first, as in the literature, a definition of a partial play. The partial play in B-M game for the player A (B) is a finite sequence of sets {Γ0 ,Γ1 , ...,Γn−1 ,Γn } ⊂ Geπ such that Γ0 ⊃ Γ1 ⊃ Γ2 ⊃ Γ3 ⊃ ... ⊃ Γn−1 ⊃ Γn

7. Generalized (topological) metric space. From nowhere density. . .

99

and Γn was chosen by the player B (A ). To facilitate of definitions and formulas let us assume that, if the player A (B) chooses first then the sequence {Γ0 } is the partial play in B-M for the player A (B). The set of all partial plays in B-M game for the player A (B) will be denoted by P(A ) (P(B)). The strategy in B-M game for the player A (B) is a function η : P(A ) → Geπ (η : P(B) → Geπ ), such that η({Γ0 ,Γ1 , ...,Γn−1 ,Γn }) ⊂ Γn . In the theory of infinite games, the existence of winning strategy for a given player is one of the most essential questions. It may be dependent (except of the case of determined games) on properties of some mathematical objects. Roughly speaking a winning strategy is a possibility of such activity of the player that it determines its victory independently of the reaction of the other player. So, let us start with the definitions. We shall say that a strategy η : P(A ) → Geπ (η : P(B) → Geπ ) is winning in B-M game for the player A (B) if for any decreasing sequence of sets {Γn }n∈N0 ⊂ Geπ with the property: for any i ∈ N, if {Γ0 ,Γ1 , ...,Γi−1 } ∈ P(A ) ({Γ0 ,Γ1 , ...,Γi−1 } ∈ P(B)) then Γi = η({Γ0 ,Γ1 , ...,Γi−1 }) we have that

∞ T

Γn 6= 0/ (

n=0

∞ T

Γn = 0). /

n=0

Now, we will present two theorems connected with B-M game in GMS (X, πX ) (let us recall that we consider here exclusively GMS (X, πX ) having a perfect kernel π0 ). Although the theorems have their analogous in the case of classical spaces, the proofs differ significantly from the earlier results. Theorem 7.10 (Theorem 5.1 [19]). A space (X, πX ) is a Baire space if and only if there is no winning strategy in B-M game for the player A whenever A chooses first. Theorem 7.11 (Proposition 5.2 [19]). There is no winning strategy in B-M game for the player B.

7.3.2 An S-F game This part contains considerations connected with original game described in [19]. This game does not have its equivalent in earlier research. Obviously, in the literature one can find infinite games connected with set valued function, e.g. [1] (the notion topological game has been introduced there) or [26]. However, they are of different character then the game presented

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below. Additional thing that distinguishes this game is the fact that there are three players taking part in it, however it is not connected with the number of its authors. Nevertheless, in the below notations the authors’ names will be reflected. So, let us fix GMS (X, πX ) which has a perfect kernel and assume that three players take part in our game: K , L and P (the first letters of the names of the authors of this chapter), who choose alternately set valued functions according to fixed rules. Player K either choose first or after the player P. Player P either choose first or after the player L . Player L either choose first or after the player K . In this game, players use a special kind of set valued functions i. e. set valued functions having a fixed set. We call U ∈ GeπX a fixed set for a set valued function F : X ( X if U ⊂ F(x) for each x ∈ U. The family of all fixed sets for a set valued function F will be denoted by F(F). The symbol FIXF (X) stands for the family of all set valued functions F : X ( X such that F(F) 6= 0. / We start by putting F0 (x) = X for x ∈ X and fixing the first player. The game follows by the rules The first player chooses F1 ∈ FIXF (X), such that F1 @ F0 . The second player chooses F2 ∈ FIXF (X), such that F2 @ F1 . The third player chooses F3 ∈ FIXF (X), such that F3 @ F2 . The first player chooses F4 ∈ FIXF (X), such that F4 @ F3 . The second player chooses F5 ∈ FIXF (X), such that F5 @ F4 , etc. In view of the fact that players in this game choose set valued functions, we will call it a set valued function game (S-F game for short). In order to make further notation clear, we will sometimes use upper index K (P or L ) to denote a set valued function chosen by K (P or L ), i.e. the notation Fn = FK n means that the set valued function Fn was chosen by the player K . Now, we need to define the rules of wins for individual players. The player K wins in S-F game if the sequence {Fn }n∈N0 is s-convergent to a set valued function F ∈ FIXF (X).

The player L wins in S-F game if the sequence {Fn }n∈N0 is s-convergent to a set valued function F 6∈ FIXF (X).

The player P wins in S-F game if the sequence {Fn }n∈N0 is not s-convergent. a a

Similarly to the case of B-M game, we will define now the notions of a partial play and strategy in S-F game.

7. Generalized (topological) metric space. From nowhere density. . .

101

The partial play in S-F game for the player K (L or P) is a finite sequence of set valued functions {F0 , F1 , ..., Fn } ⊂ FIXF (X) such that Fn @ Fn−1 @ ... @ F1 @ F0 K L and Fn = FP n (Fn = Fn or Fn = Fn ). Moreover, if the player K (L or P) chooses first then the sequence {F0 } is the partial play in S-F game for the player K (L or P). The set of all partial plays in S-F game for the player K (L or P) will be denoted by S(K ) (S(L ) or S(P)). The strategy in S-F game for the player K (L or P) is a function ξ : S(K ) → FIXF (X) (ξ : S(L ) → FIXF (X) or ξ : S(P) → FIXF (X)), such that ξ ({F0 , F1 , ..., Fn }) @ Fn . We shall say that a strategy ξ : S(K ) → FIXF (X) (ξ : S(L ) → FIXF (X) or ξ : S(P) → FIXF (X)) is winning in S-F game for the player K (L or P) if for any decreasing sequence of set valued functions {Fn }n∈N0 ⊂ FIXF (X) with the property

for any i ∈ N if {F0 , F1 , ..., Fi−1 } ∈ S(K ) ({F0 , F1 , ..., Fi−1 } ∈ S(L ) or {F0 , F1 , ..., Fi−1 } ∈ S(P)) then Fi = ξ ({F0 , F1 , ..., Fi−1 }) we have that the sequence {Fn }n∈N0 is s-convergent to F ∈ FIXF (X) ({Fn }n∈N0 is s-convergent to F 6∈ FIXF (X) or {Fn }n∈N0 is not s-convergent). Let us begin with the situation when player K chooses first in S-F game. Then we have: Theorem 7.12 (Theorem 5.6 [19]). If there is an isolated point in the space (X, πX ) and the player K chooses first in S-F game then the player K has a winning strategy in S-F game. The natural consequence is considering the situation when player K does not choose first in S-F game. Then: Theorem 7.13 (Theorem 5.5 [19]). If the player K does not choose first in S-F game and K has a winning strategy in S-F game then (X, πX ) is a strong Baire space. In the previous theorem we have assumed that K does not choose first in S-F game and K has a winning strategy. In consequence, the next question arises: When does player K have a winning strategy? Theorem 7.14 (Theorem 5.4 [19]). If the player K does not choose first in S-F game then K has a winning strategy in S-F game if and only if the set of all isolated points of (X, πX ) is dense.

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Finally, we will refer to considerations connected with players L and P. Theorem 7.15 (Theorem 5.3 (ii) [19]). If the player L chooses first in S-F game and there is no winning strategy in SFG for the player L then (X, πX ) is a Baire space. Theorem 7.16 (Theorem 5.3 (i) [19]). If (X, πX ) is a Baire space with no isolated points then there is no winning strategy in S-F game for the player P.

7.4 Recent results The research concerning GMS are continued by the authors of this chapter in [25]. In view of the fact that this paper has not been published yet, we will only signal a piece of it. A notion of a base consisting of metrics, introduced in this paper, is particularly important. This notion has a close relationship with the notion of kernel presented earlier in this chapter. However, we will concentrate here on the generalization of the unit interval. First, let us define some kind of metric. Let A ⊂ [0, 1]. From now on, the symbol σA stands for the following metric (called almost natural metric): ( ρE (x, y) if x, y ∈ A or x, y ∈ / A, σA (x, y) = 1 otherwise. Obviously, in the above definition we can consider any nonempty set A. However, if we take into account an interval [a, b] ⊂ [0, 1], we obtain the metric σ[a,b] having some special properties (writing σ[a,b] we assume that a < b). Clearly, in this case we can obtain for example the following situation: σ[a,b] (x, y) = |x − y| 0

x

σ[a,b] (z,t) = |z − t| y

a

z

t

b=1

σ[a,b] (y, z) = 1

In paper [25], generalized metric spaces connected with metrics of the form σ[a,b] were investigated. Such kind of GMS is called π-unit interval. For example we have that GMS ([0, 1], π) is a π-unit interval if π consists of a finite number of almost natural metrics of the form σ[a,b] . From our point of view, the following theorem is important.

7. Generalized (topological) metric space. From nowhere density. . .

103

Theorem 7.17 ([25]). Every π-unit interval is a Baire space.

References [1] C. Berge, Topological games with perfect information, in: Contributions to the theory of games, Vol. III, Annals of Math. Studies 39, Princeton University Press, Princeton 1957, 165-178. [2] E. Borel, La théorie du jeu et les équations intégrales à noyau symétrque, C. R. Acad. Sci. Paris 173 (1921), 1304-1308; English transl.: The theory of play and integral equations with skew symmetric kernels, Econometrica 21 (1953), 97-100. [3] E. Borel, Sur les jeux où interviennent l’hasared et l’habileté des joueurs, in Theories des probabilities, Herman, Paris (1924), 204-224; English transl.: On games that involve chance and the skill of the players, Econometrica 21 (1953), 101-115. [4] E. Borel, Sur les systèmes de formes linéaires à déterminant symétrique gauche et la théorie générale du jeu, C. R. Acad. Sci. Paris 184 (1927), 52-54; English transl.: On systems of linear forms of skew symmetric determinant and general theory of play, Econometrica 21 (1953), 116-117. [5] J. Borsík, Generalized oscillations for generalized continities, Tatra Mt. Math. Publ. 49 (2011), 119-125. [6] A. Blass, Complexity of winning strategies, Discrete Math. 3 (1972), 295-300. [7] J. Burgess, D. Miller, Remarks on invariant descriptive set theory, Fund. Math. 90 (1975), 53-75. [8] J. P. Burgess, R. A. Lockhart, Classical hierarchies from a modern standpoint, Part III: BP-sets, Fund. Math. 115 (1983), 107-118. [9] Á. Császár, Generalized open sets, Acta Math. Hungar. 75 (1-2) (1997), 65-87. [10] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (4) (2002), 351-357. [11] Á. Császár, γ-connected sets, Acta Math. Hungar. 101 (4) (2003), 273-279. [12] Á. Császár, Separation axioms for generalized topologies, Acta Math. Hungar. 104 (1-2) (2004), 63-69. [13] Á. Császár, Modification of generalized topologies via hereditary classes, Acta Math. Hungar. 115 (1-2) (2007), 29-36. [14] Á. Császár, Normal generalized topologies, Acta Math. Hungar. 115 (4) (2007), 309313. [15] Á. Császár, Product of generalized topologies, Acta Math. Hungar. 123 (1-2) (2009), 127-132. [16] A. Ehrenfeucht, An application of games to the completeness problem for formalized theories, Fund. Math. 49 (1961), 129-141. [17] J. P. Jones, Recursive undecidability - an exposition, Amer. Math. Monthly 81 (1974), 724-738. [18] J. P. Jones, Some undecidable determined games, Internat. J. Game Theory 11 (1982), 63-70. [19] E. Korczak-Kubiak, A. Loranty, R. J. Pawlak, Baire generalized topological spaces, generalized metric spaces and infinite games, Acta Math. Hungar. 140 (3) (2013), 203-231.

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[20] A. L. Lachlan, On some games which are relevant to the theory of recursively enumerable sets, Ann. of Math. 91 (1970), 291-310. [21] J. Li, Generalized topologies generated by subbases, Acta Math. Hungar. 114 (1-2) (2007), 1-12. [22] A. Loranty, R. J. Pawlak, On the transitivity of multifunctions and density of orbits in generalized topological spaces, Acta Math. Hungar. 135(1-2) (2012), 56-66. [23] J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928), 295320. [24] R. J. Pawlak, A. Loranty, The generalized entropy in the generalized topological spaces, Topology Appl. 159 (2012), 1734-1742. [25] R. J. Pawlak, A. Loranty, E. Korczak-Kubiak On stronger and weaker forms of continuity in GTS - properties and dynamics, Topology and its Applications (in print). [26] J. P. Revalski, The Banach-Mazur Game: History and Recent Developments, Institute of Mathematics and Informatics Bulgarian Academy of Sciences, Seminar notes, Pointe-a-Pitre, Guadeloupe, France, 2003-2004, http://www1.univag.fr/aoc/activite/revalski/ [27] M. S. Sarsak, Weak separation axioms in generalized topological spaces, Acta Math. Hungar. 131 (1-2) (2011), 110-121. [28] H. Steinhaus, Definicje potrzebne do teorii gier i po´scigu, (in Polish), Złota My´sl Akademicka, Lwów, Vol. 1, No. 1, (1925), 13-14; English transl.: Definitions for a theory of games and pursuits, Naval Res. Logist. Quart. 7 (1960), 105-108. [29] R. Telgársky, Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math. 17 (2) (1987), 227-276. [30] S. M. Ulam, The Scottish Book, Los Alamos, CA, 1977. [31] C. E. M. Yates, Prioric games and minimal degrees below 0, Fund. Math 82 (1974), 217-237. [32] C. E. M. Yates, Banach-Mazur games, comeager sets and degress of unsolvability, Math. Proc. Cambridge Philos. Soc. 79 (1976), 195-220.

E WA KORCZAK -K UBIAK Faculty of Mathematics and Computer Science, Łód´z University Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

A NNA L ORANTY Faculty of Mathematics and Computer Science, Łód´z University Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

RYSZARD J. PAWLAK Faculty of Mathematics and Computer Science, Łód´z University Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

Chapter 8

Path continuity connected with density and porosity

STANISŁAW KOWALCZYK AND MAŁGORZATA TUROWSKA

2010 Mathematics Subject Classification: 54C30, 26A15, 54C08. Key words and phrases: path continuity, density, porosity, v-porosity, ρ-upper continuity, [λ , ρ]-continuity, porouscontinuity, v-porouscontinuity, maximal additive class, maximal multiplicative class.

8.1 Preliminary The notion of path continuity appeared in the theory of real functions at the beginning of XXth century. Path continuity means that at every x ∈ R a family Bx of subsets of R, „big” in some sense near x and containing x, is defined and f : R → R is continuous at x0 relative to {Bx : x ∈ R} if there exists E ∈ Bx0 such that f |E is continuous at x0 . Taking different families of sets we get different kinds of path continuity. The first and most important, without a doubt, is an approximate continuity, where as Bx we take a family of measurable sets for which x is a point of density. In [6] a lot types of path continuity is described, including preponderant continuity, qualitative continuity, Császár’s continuity and congruent continuity. Applying system of paths other properties of functions can be generalized, including derivative. Deep studying of generalized path derivatives can be found in [7]. In [27] majority of notions of real analysis is defined based on system of paths. In [1, 2, 3] basic properties of path continuity

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were studied. Other problems connected with path continuity were studied in [4, 11, 15, 20, 21, 22]. In our paper we concentrate on path continuity related to the notions of porosity and lower and upper density. We will use standard notations. R denotes the set of reals and N denotes the set of positive integers. For f : R → R, let C ( f ), D( f ), Q( f ) and Dap ( f ) denote the sets of points at which f is continuous, discontinuous, quasicontinuous and approximately discontinuous, respectively. The symbols C , Q and A stand for the sets of continuous functions, quasicontinuous functions and approximately continuous functions, respectively. Moreover, C ± is the set of all functions f : R → R such that at every x ∈ R, f is continuous from the left or from the right (obviously C $ C ± ). Finally, µ denotes the Lebesgue measure on R and N f = {x : f (x) = 0}.

8.2 Path continuity connected with density Path continuities defined via lower and upper density were introduced in [7, 13, 23]. They are generalizations of preponderant density which was defined by Denjoy in [8]. Let E be a measurable subset of R and let x ∈ R. The numbers d + (E, x) = lim inf + t→0

µ(E ∩ [x, x + t]) µ(E ∩ [x, x + t]) + and d (E, x) = lim sup t t t→0+

are called the right lower density of E at x and right upper density of E at x, respectively. The left lower and upper densities of E at x are defined analogously. If  + − d + (E, x) = d (E, x) d − (E, x) = d (E, x) then we call this number the right density (left density) of E at x and denote it by d + (E, x) (d − (E, x)). The numbers d(E, x) = lim sup t→0+ k→0+

µ(E ∩ [x − t, x + k]) µ(E ∩ [x − t, x + k]) , d(E, x) = lim inf k +t k +t t→0+ + k→0

are called the upper and lower density of E at x, respectively. If d(E, x) = d(E, x), we call this number the density of E at x and denote it by d(E, x).

8. Path continuity connected with density and porosity

107

Let us observe that +



d(E, x) = max {d (E, x), d (E, x)} and d(E, x) = min {d + (E, x), d − (E, x)}. Moreover, it is clear that +

d (E, x) + d + (R \ E, x) = 1

and



d (E, x) + d − (R \ E, x) = 1.

Definition 8.1. [13, 14] Let E be a measurable subset of R. Let x ∈ R and 0 < ρ ≤ 1. We say that the point x is a point of ρ-type upper density of E if d(E, x) > ρ when ρ < 1 or if d(E, x) = 1 when ρ = 1. Definition 8.2. [13, 14] Let ρ ∈ (0, 1]. A real-valued function f defined on R is called ρ-upper continuous at x provided that there is a measurable set E ⊂ R such that the point x is a point of ρ-type upper density of E, x ∈ E and f |E is continuous at x. If f is ρ-upper continuous at every point of R we say that f is ρ-upper continuous. We will denote the class of all ρ-upper continuous functions by U C ρ . Corollary 8.1. Let 0 < ρ < ρ1 ≤ 1, f : R → R and x ∈ R. If f is ρ1 -upper continuous at x, then f is ρ-upper continuous at x. Theorem 8.1. [13] Let ρ, ρ1 ∈ (0, 1]. Then U C ρ ⊂ U C ρ1 if and only if ρ1 ≤ ρ. Moreover, if ρ1 < ρ then U C ρ $ U C ρ1 . In an obvious way we define one-sided ρ-upper continuity, and f is ρ-upper continuous at x if and only if it is ρ-upper continuous at x from the right or from the left. Moreover, if a function is continuous from the right or from the left at some point then it is ρ-upper continuous at this point. Next definitions are based on the notion of preponderant continuity in O’Malley sense [25, 12]. Definition 8.3. [17] Let ρ ∈ (0, 1). A point x is said to be a point of ρ-type upper density in O’Malley sense of a measurable set E if for each ε > 0 there exists y ∈ (x − ε, x + ε) such that for the closed interval J with endpoints x and y the inequality µ(E∩J) |x−y| > ρ holds. Definition 8.4. [17] Let ρ ∈ (0, 1). A function f : R → R is said to be ρ-upper continuous in O’Malley sense at x ∈ R ( f is O’Malley ρ-upper continuous at x, in abbreviation) if there exists a measurable set E ⊂ R containing x such that x is a point of ρ-type upper density in O’Malley sense of E and f |E is continuous at x. A function f is said to be O’Malley ρ-upper continuous if it is O’Malley ρ-upper continuous at each point of R.

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We denote the class of O’Malley ρ-upper continuous functions by OU C ρ . Corollary 8.2. Let 0 < ρ < ρ1 < 1, f : R → R and x ∈ R. If f is O’Malley ρ1 -upper continuous at x, then f is O’Malley ρ-upper continuous at x. Theorem 8.2. [17] Let ρ, ρ1 ∈ (0, 1). Then OU C ρ ⊂ OU C ρ1 if and only if ρ1 ≤ ρ. Moreover, if ρ1 < ρ then OU C ρ $ OU C ρ1 . In an obvious way we define one-sided O’Malley ρ-upper continuity and f is O’Malley ρ-upper continuous at x if and only if it is O’Malley ρ-upper continuous at x from the right or from the left. Moreover, if a function is continuous from the right or from the left at some point then it is O’Malley ρ-upper continuous at this point. Now, we give next definition of path continuity connected with lower and upper density simultaneously. Definition 8.5. [23] Let E be a measurable subset of R and let 0 < λ ≤ ρ ≤ 1. We say that a point x ∈ R is a point of [λ , ρ]-density of E if d(E, x) > λ and d(E, x) > ρ for ρ < 1 or if d(E, x) > λ and d(E, x) = 1 for ρ = 1 and λ < 1 or if d(E, x) = 1 for λ = 1. Definition 8.6. [23] Let 0 < λ ≤ ρ ≤ 1. A function f : R → R is called [λ , ρ]continuous at x ∈ R, provided that there is a measurable set E ⊂ R such that x is a point of [λ , ρ]-density of E, x ∈ E and f |E is continuous at x. If f is [λ , ρ]-continuous at each point of R, we say that f is [λ , ρ]-continuous. We will denote the class of all [λ , ρ]-continuous functions by C[λ ,ρ] . Remark 8.1. C[1,1] is the class of approximately continuous functions and C[ 1 , 1 ] is the class of preponderantly continuous functions. Approximately con2 2 tinuous functions and preponderantly continuous functions were studied much earlier, see [8, 28]. Since approximately continuous functions were very widely studied, we will consider only classes C[λ ,ρ] , where λ < 1. Corollary 8.3. [23] Let 0 < λ ≤ ρ ≤ 1, 0 < λ1 ≤ ρ1 ≤ 1, ρ ≤ ρ1 , λ ≤ λ1 , f : R → R and x ∈ R. If f is [λ1 , ρ1 ]-continuous at x, then f is [λ , ρ]-continuous at x. Theorem 8.3. [23] Let 0 < λ ≤ ρ ≤ 1, 0 < λ1 ≤ ρ1 ≤ 1. Then C[λ1 ,ρ1 ] ⊂ C[λ ,ρ] if and only if ρ ≤ ρ1 and λ ≤ λ1 . If, moreover, if ρ < ρ1 or λ < λ1 then C[λ1 ,ρ1 ] $ C[λ ,ρ] .

8. Path continuity connected with density and porosity

109

Clearly, if a function is continuous at some point then it is [λ , ρ]-continuous at this point. It is worth to mention that functions which are continuous from the left or from the right at some point need not be [λ , ρ]-continuous at this point. In an obvious way we define one-sided [λ , ρ]-continuity. Then, if f is [λ , ρ]-continuous at x then f is [λ , ρ]-continuous at x from the right or from the left. On the other hand, if f is [λ , ρ]-continuous at x from the right and from the left then f is [λ , ρ]-continuous at x. Example 8.1. Take any 0 < λ < ρ ≤ 1. Let ([an , bn ])n∈N be a sequence of  ∞  + S closed intervals such that 0 < . . . bn+1 < an < bn . . . and d [an , bn ], 0 = n=1  ∞  ∞ S ρ+λ + S d [an , bn ], 0 = 2 . Let I = [an , bn ]. Let ([cn , dn ])n∈N be a sequence n=1

n=1

of closed pairwise disjoint intervals such that [an , bn ] ⊂ (cn , dn ) for n ≥ 1 and  ∞  + S ([cn , dn ] \ [an , bn ]), 0 = 0. Define f : R → R by d n=1

  0, x ∈ (−∞, 0] ∪ I,    ∞ S x ∈ [d1 , ∞] ∪ [dn+1 , cn ], f (x) = 1,  n=1   linear on each interval [c , a ] and [b , d ], n n n n

n ≥ 1.

Obviously, f is continuous at each point except 0. Let E = (−∞, 0] ∪ I. Then − f |E is constant, 0 ∈ E, d(E, 0) = d (E, 0) = 1 and d(E, 0) = d + (E, 0) = ρ+λ 2 > λ . Thus f is [λ , ρ]-continuous at 0 and f ∈ C[λ ,ρ] . On the other hand,  ∞  + + S + d ({x : | f (x) − f (0)| < 21 }, 0) ≤ d [cn , dn ], 0 = d (I, 0) = ρ+λ 2 < ρ. n=1

Thus f is not [λ , ρ]-continuous at 0 from the right. Example 8.2. Take any 0 < λ < ρ ≤ 1. Define f : R → R by ( 0, x ∈ [0, ∞), f (x) = 1, x ∈ (−∞, 0). Clearly, f is continuous at each point except 0 and is continuous from the right + at 0. Let E = [0, ∞). Then f |E is constant, 0 ∈ E and d (E, 0) = d + (E, 0) = 1. Thus f is [λ , ρ]-continuous at 0 from the right. On the other hand d({x : | f (x) − f (0)| < 21 }, 0) = d − ({x : | f (x) − f (0)| < 21 }, 0) = 0. Thus f is not [λ , ρ]-continuous at 0.

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8.3 Porouscontinuity and v-porouscontinuity In [5] J. Borsík and J. Holos introduced path continuity connected with the notion of porosity. These notions were generalized in [19]. For a set A ⊂ R and an interval I ⊂ R let Λ (A, I) denote the length of the largest open subinterval of I having an empty intersection with A. Let x ∈ R. Then, according to [5, 29, 9], the right-porosity of the set A at x is defined as p+ (A, x) = lim sup h→0+

Λ (A, (x, x + h)) , h

the left-porosity of the set A at x is defined as p− (A, x) = lim sup h→0+

Λ (A, (x − h, x)) , h

and the porosity of A at x is defined as  p(A, x) = max p− (A, x), p+ (A, x) . Definition 8.7. [5] A point x ∈ R will be called a point of πr -density of the set A ⊂ R for r ∈ [0, 1) (µr -density of a set A for r ∈ (0, 1]) if p(R \ A, x) > r (p(R \ A, x) ≥ r). Definition 8.8. [5] Let r ∈ [0, 1). The function f : R → R will be called • Pr -continuous at x if there exists a set A ⊂ R such that x ∈ A, x is a point of πr -density of A and f |A is continuous at x. • Sr -continuous at x if for each ε > 0 there exists a set A ⊂ R such that x ∈ A, x is a point of πr -density of A and f (A) ⊂ ( f (x) − ε, f (x) + ε). Let r ∈ (0, 1]. The function f : R → R will be called • Mr -continuous x if there exists a set A ⊂ R such that x ∈ A, x is a point of µr -density of A and f |A is continuous at x; • Nr -continuous x if for each ε > 0 there exists a set A ⊂ R such that x ∈ A, x is a point of µr -density of A and f (A) ⊂ ( f (x) − ε, f (x) + ε). All of these functions are called porouscontinuous functions. Symbols Pr ( f ), Sr ( f ), Mr ( f ) and Nr ( f ) denote the set of all points at which f is Pr -continuous, Sr -continuous, Mr -continuous and Nr -continuous, respectively. In [5] the equality Mr ( f ) = Nr ( f ) for every f and every r ∈ (0, 1] was proved.

8. Path continuity connected with density and porosity

111

Theorem 8.4. [18] Let r ∈ [0, 1), x ∈ R and f : R → R. Then x ∈ Pr ( f ) if and only if lim+ p(R \ {t : | f (x) − f (t)| < ε}, x) > r. ε→0

In an obvious way we define one-sided porouscontinuity, and f is porouscontinuous at x if and only if it is porouscontinuous at x from the right or from the left. Moreover, if f is continuous from the right or from the left at some x then f is porouscontinuous at this point. Theorem 8.5. [5] Let 0 < r < s < 1 and f : R → R. Then C ( f )⊂ M1 ( f )⊂ Ps ( f )⊂ Ss ( f )⊂ Ms ( f )⊂ Pr ( f )⊂ P0 ( f )⊂ S0 ( f )⊂ Q( f ). Following [5], we introduce the following denotations: • for r ∈ (0, 1] let Mr = { f : Mr ( f ) = R}, • for r ∈ [0, 1) let Pr = { f : Pr ( f ) = R} and Sr = { f : Sr ( f ) = R}. Theorem 8.6. [5] Let 0 < r < s < 1. Then C ⊂ M1 ⊂ Ps ⊂ Ss ⊂ Ms ⊂ Pr ⊂ Mr ⊂ P0 ⊂ S0 ⊂ Q. All inclusions are proper. In [18] these results are improved. Let us consider the topology of uniform convergence, which is generated by the metric θ ( f , g) = min {1, sup {| f (x) − g(x)| : x ∈ R}} in the space of all functions from R to R. Then we can studied topological structure of presented inclusions. Theorem 8.7. [18] (1) (2) (3) (4) (5) (6)

C ± is nowhere dense and closed in M1 . For s ∈ [0, 1), M1 is nowhere dense and closed in Ps . For s ∈ (0, 1), Ss is nowhere dense and closed in Ms . For 0 ≤ r < s ≤ 1, Ms is nowhere dense and closed in Pr . For r ∈ (0, 1], Mr is nowhere dense and closed in P0 . S0 is nowhere dense and closed in Q.

Example 8.3. [18] Let F be any subfamily of Q. Take an h ∈ F and q ∈ [0, 1). Since h ∈ Q, we have C (h) 6= 0. / Let x0 ∈ C (h). There exist ε > 0 and δ > 0 such that |h(x) − h(x0 )|
0, left v-porous at x if vp− (A, x) > 0 and v-porous at x if vp(A, x) > 0. It is clear, that A is v-porous at x if and only if it is v-porous from the right or from the left at x. The set A is called v-porous if A is v-porous at each point x ∈ A. In [29] Zajíˇcek investigate similar notion, which we remind. Let X be a metric space. The open ball with the center x ∈ X and with the radius R will be denoted by B(x, R). Let M ⊂ X, x ∈ X and R > 0. Then, according to [29], by γ(x, R, M), we denote the supremum of the set of all r > 0 for which there exists z ∈ X such that B(z, r) ⊂ B(x, R) \ M. The number 2 lim sup γ(x,R,M) is R R→0+

called the porosity of M at x. We say that the set M is very porous at x if 2γ(x,R,M) > 0. lim inf R + R→0

The definition of v-porosity of the set is very similar to the definition of very porosity in the space R with the euclidian metric, but they are not equivalent. Corollary 8.4. Let A ⊂ R, x ∈ R. If the set A is v-porous at x then A is very porous at x. The next example shows that the converse implication is not true. i h i ∞ h S 1 1 1 1 Example 8.4. [19] Let A = , ∪ − , − (2n)! (2n−1)! (2n)! (2n+1)! . Then n=1

Λ (A, (0, h)) vp (A, 0) = lim inf ≤ lim n→∞ h h→0+ +

1 (2n)!

1 − (2n+1)!

1 (2n−1)!

1 n→∞ 2n+1

= lim

=0

and vp− (A, 0) = lim inf + h→0

1 1 − (2n+2)! + (2n+1)! Λ (A, (−h, 0)) 1 ≤ lim = lim 2n+2 = 0. 1 n→∞ n→∞ h (2n)!

Hence vp(A, 0) = 0.  1 2n+1 Denote tn = n!1 + n!1 − (n+1)! = (n+1)! for each n ∈ N. Let h ∈ (0, 1). There h  1 1 exists n ∈ N such that h ∈ (2n+1)! , (2n−1)! .

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If h ∈

h

1 (2n+1)! ,t2n+1



  1 1 then − (2n+1)! , − (2n+2)! ∩ A = 0/ and

2γ(0, h, A) ≥ h

1 (2n+1)!

1 − (2n+2)!

t2n+1

=

2n+1 (2n+2)! 4n+3 (2n+2)!

=

2n + 1 . 4n + 3

 h   1 1 , h ∩ A = 0/ and If h ∈ t2n+1 , (2n)! then (2n+1)! 1 1 t2n+1 − (2n+1)! 2γ(0, h, A) h − (2n+1)! ≥ ≥ = h h t2n+1

If h ∈

h

1 (2n)! ,t2n



then



2γ(0, h, A) ≥ h

1 1 (2n+1)! , (2n)!

1 (2n)!



1 − (2n+1)!

h

2n+1 (2n+1)! 4n+3 (2n+2)!

=

2n + 1 . 4n + 3

∩ A = 0/ and



2n (2n+1)!

t2n

=

2n (2n+1)! 4n+1 (2n+1)!

=

2n . 4n + 1

h    1 1 Finally, if h ∈ t2n , (2n−1)! then −h, − (2n)! ∩ A = 0/ and 1 1 t2n − (2n)! 2γ(0, h, A) h − (2n)! ≥ ≥ = h h t2n 2γ(0,h,A) = Therefore lim inf h + h→0

1 2

2n (2n+1)! 4n+1 (2n+1)!

=

2n . 4n + 1

and A is very porous at 0.

Moreover, +

p(A, 0) ≥ p (A, 0) ≥ lim

n→∞

   1 Λ A, 0, (2n)! 1 (2n)!

= lim

n→∞

1 (2n)!

1 − (2n+1)! 1 (2n)!

= 1.

Definition 8.9. [19] A point x ∈ R will be called a point of vπr -density of a set A ⊂ R for r ∈ [0, 1) (vµr -density of a set A for r ∈ (0, 1]) if vp(R \ A, x) > r (vp(R \ A, x) ≥ r). In an obvious way, we may define one-sided vπr -densities and vµr -densities. Definition 8.10. [19] Let r ∈ [0, 1). The function f : R → R will be called • vPr -continuous at x if there exists a set A ⊂ R such that x ∈ A, x is a point of vπr -density of A and f |A is continuous at x; • vSr -continuous at x if for each ε > 0 there exists a set A ⊂ R such that x ∈ A, x is a point of vπr -density of A and f (A) ⊂ ( f (x) − ε, f (x) + ε).

8. Path continuity connected with density and porosity

115

Let r ∈ (0, 1]. The function f : R → R will be called • vMr -continuous at x if there exists a set A ⊂ R such that x ∈ A, x is a point of vµr -density of A and f |A is continuous at x; • vNr -continuous at x if for each ε > 0 there exists a set A ⊂ R such that x ∈ A, x is a point of vµr -density of A and f (A) ⊂ ( f (x) − ε, f (x) + ε). The symbols vPr ( f ), vSr ( f ), vMr ( f ) and vNr ( f ) denote the sets of all points at which the function f is vPr -continuous, vSr -continuous, vMr continuous and vNr -continuous, respectively. These sets will be called the sets of v-porouscontinuity points of the function f . In an obvious way, we may define one sided v-porouscontinuity of the function f : R → R at a point. Symbols vPr+ ( f ), (vPr− ( f )), vSr+ ( f ), (vSr− ( f )), vMr+ ( f ), (vMr− ( f )) and vNr + ( f ), (vNr − ( f )) will denote the sets of all points at which the function f is vPr -continuous from the right (from the left), vSr -continuous from the right (from the left), vMr -continuous from the right (from the left) and vNr -continuous from the right (from the left), respectively. Obviously, vPr ( f ) = vPr+ ( f ) ∪ vPr− ( f ), vSr ( f ) = vSr+ ( f ) ∪ vSr− ( f ) and vMr ( f ) = vMr+ ( f ) ∪ vMr− ( f ). If f is continuous from the right or from the left at x then x ∈ vPr ( f ) ∩ vSr ( f ) ∩ vMs ( f ) for every r ∈ [0, 1) and every s ∈ (0, 1]. Theorem 8.9. [19] Let A ⊂ R, x ∈ A. If there exists a decreasing sequence (xn )n∈N ⊂ A converging to x, then vp+ (A, x) ≤ 21 . Corollary 8.5. vPr ( f ) = vSr ( f ) = C ( f ) for each r ∈ [ 12 , 1). Similarly, vMr ( f ) = vNr ( f ) = C ( f ) for each r ∈ ( 21 , 1]. Theorem 8.10. [19] Let r ∈ (0, 21 ] and f : R → R. Then vMr ( f ) = vNr ( f ). We introduce the following denotations: • for r ∈ (0, 1] let vMr = { f : vMr ( f ) = R}, • for r ∈ [0, 1) let vPr = { f : vPr ( f ) = R} and vSr = { f : vSr ( f ) = R}. Corollary 8.6. vMr = vNr for every r ∈ (0, 12 ]. Theorem 8.11. [19] Let r ∈ [0, 12 ), x ∈ R and f : R → R. Then x ∈ vPr ( f ) if and only if lim+ vp(R \ {t : | f (x) − f (t)| < ε}, x) > r. ε→0

Theorem 8.12. [19] Let 0 < r < s
0 and = r for every k ∈ N. x xk   ∞ S + Hence d In , 0 = r. Let {[cn , dn ]}n≥1 be a sequence of pairwise disjoint

n=1

8. Path continuity connected with density and porosity

117

closed intervals such that [yn , xn ] ⊂ (cn , dn ) for every n ≥ 1 and, moreover,  ∞  + S d ([cn , dn ] \ [yn , xn ]), 0 = 0. Finally, let Jn = [cn , dn ] for every n ≥ 1. Den=1

fine f : R → R by  ∞ S  0, x ∈ {0} ∪ In ,    n=1 ∞ f (x) = 1, x ∈ (−∞, 0) ∪ S [dn+1 , cn ] ∪ [d1 , ∞),    n=1  linear on each interval [cn , yn ], [xn , dn ], n = 1, 2, . . . . It is easily seen that f is continuous at every point except 0. Since d

+

   ∞  + [ {x : | f (x)| < 1}, 0 ≤ d Jn , 0 ≤ n=1

≤d

+

∞ [ n=1

  ∞  + [ In , 0 + d (Jn \ In ), 0 = r, n=1

we conclude that f is not r-upper continuous at 0. Hence f ∈ / U C r. On the other hand, for every ε > 0 and for every y > 0 we have n ∞ o [ µ x ∈ [0, y] \ In : | f (x)| < ε > 0. n=1

Hence n o  ∞ S   µ x∈ In : | f (x)| < ε ∩ [0, xk ] µ x : | f (x)| < ε ∩ [0, xk ] n=1 = + xk xk n o  ∞ S µ x∈ / In : | f (x)| < ε ∩ [0, xk ] n=1 + = xk n o  ∞ S µ x∈ / In : | f (x)| < ε ∩ [0, xk ] n=1 = r+ > r. xk Since lim xk = 0, we conclude that f is r-upper continuous in O’Malley sense k→∞

at 0 and f ∈ OU C r . Moreover, it is obvious that f ∈ U C r0 and f ∈ / OU C r1 . Since d({x : | f (x) − f (0)| < ε}, 0) = d − ({x : | f (x) − f (0)| < ε}, 0) = 0 for ε ∈ (0, 1), we conclude that f ∈ / C[r0 ,r0 ] .

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This example shows that inclusions in Theorem 8.15, Theorem 8.16 and Theorem 8.17 are proper. Example 8.7. Let A be the set from Example 8.4. Then the characteristic function of the set R \ A belongs to M1 and does not belong to vS0 . Theorem 8.20. U C ρ1 6⊂ C[λ ,ρ] for any ρ, ρ1 ∈ (0, 1], λ ∈ (0, ρ]. Proof. Let χ(−∞,0] be the characteristic function of (−∞, 0]. Then f ∈ U C ρ1 but f ∈ / C[λ ,ρ] for any ρ, ρ1 ∈ (0, 1], λ ∈ (0, ρ]. t u Relationship between discussed classes of function for r ∈ (0, 1) and λ ∈ (0, r) can be represented in the following diagram.

r vP 1+r

r vS 1+r

Mr

$

r vM 1+r

U Cr

$

$

OU C r

%

C[λ ,r]

$

$

$

$

Sr $

$

$

Pr

C[r,r]

8.5 Properties of path continuous functions In this section we will consider measurability, connections with Baire one functions, uniform convergence, adding and multiplying of functions from defined classes. OU C r is measurable.

S

Theorem 8.21. [13] Every function from

r∈(0,1)

Applying Theorem 8.21 and inclusions showed in section 8.4, we obtain Theorem 8.22. Every function from [

(U C r ∪ Mr ) ∪

r∈(0,1]

[

(Pr ∪ Sr ) ∪

[ r∈(0, 12 ]

is measurable.

(vPr ∪ vSr )∪

r∈[0, 21 )

r∈[0,1)



[

vMr ∪

[ 0 λ.

Thus fn ∈ C[λ ,1] ⊂ C[λ ,ρ] for n ≥ 1. It is easily seen that result of addition and multiplication of functions from discussed classes of functions, in general, need not belong to these classes. Therefore we studied the following notion. Definition 8.11. Let F be a family of real functions defined on R. A set Ma (F ) = {g : R → R : ∀ f ∈F f + g ∈ F } is called the maximal additive class for F . Remark 8.2. Let f : R → R, f (x) = 0 for x ∈ R be a constant function. Clearly, if f ∈ F then Ma (F ) ⊂ F . Theorem 8.28. [16] Ma (C[λ ,ρ] ) = A for 0 < λ ≤ ρ ≤ 1. In the sequel we will need the notions of sparse sets and T ∗ -continuity. Definition 8.12. [26] A measurable set E is called sparse at a point x ∈ R from the right if for each ε > 0 there exists k > 0 such that any interval (α, β ) ⊂ (x, x + k) satisfying condition α − x < k(β − x) contains at least one point y such that µ(E ∩ (x, y)) < ε(y − x). Analogously, we can define a left-sided sparsity. A family of all measurable sets which are sparse from the right (from the left) at x we denote by S(x+) (S(x−)). Definition 8.13. [26] We say that a measurable set E is sparse at x if E ∈ S(x), where S(x) = S(x+) ∩ S(x−). Theorem 8.29. [26] Let x ∈ R and let E be a measurable subset of R. The following conditions are equivalent 1. E ∈ S(x+), + + 2. if F ⊂ R is a measurable set and d (F, x) < 1 then d (E ∪ F) < 1, + 3. if F ⊂ R is a measurable set, d + (F, x) = 0 and d (F, x) < 1 then + d + (E ∪ F, x) = 0 and d (E ∪ F, x) < 1, 4. if F ⊂ R is a measurable set and d + (F, x) = 0 then d + (E ∪ F, x) = 0.

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Stanisław Kowalczyk and Małgorzata Turowska +

Obviously, if d (E, x) = 0 then E is sparse from the right at x. In [10] it is shown that there exists a measurable set E which is sparse from the right at + + some point x and d (E, x) > 0. Moreover, if d + (E, x) > 0 or d (E, x) = 1 then E is not sparse from the right at x. According to [10] we have: Definition 8.14. [10] A real-valued function f defined on R is called T ∗ continuous if for each a ∈ R the sets {x : f (x) < a} and {x : f (x) > a} are complements of sparse sets. We denote by CT ∗ a family of T ∗ -continuous functions. Clearly, A is a proper subset of CT ∗ . Since complements of sparse sets form a topology, [10], we can define T ∗ continuity locally. Definition 8.15. [10] We say that a function f : R → R is T ∗ -continuous at x0 ∈ R if for each ε > 0 the complement of the set {x : | f (x) − f (x0 )| < ε} is sparse at x0 . Corollary 8.8. A function f : R → R is T ∗ -continuous if and only if it is T ∗ continuous at each point of R. Theorem 8.30. [14] 1. Ma (U C ρ ) = A for ρ ∈ (0, 1), 2. Ma (U C 1 ) = CT ∗ . Theorem 8.31. [17] Ma (OU C ρ ) = C for r ∈ (0, 1), Theorem 8.32. Ma (Pr ) = Ma (Sr ) = Ma (Mr ) = C for r ∈ (0, 1). Proof. Inclusions C ⊂ Ma (Pr ), C ⊂ Ma (Sr ), C ⊂ Ma (Mr ) for r ∈ (0, 1) are obvious.  2r Fix r ∈ (0, 1) and take any f : R → R, f ∈ Pr \ C and c ∈ r, 1+r . Then c 1 1 2−c = 2 −1 < 2 −1 = r. By assumption, f is discontinuous at some x0 . We c

2r r+1

may assume that x0 = 0 and f is discontinuous from the right at 0. Let ε > 0 and (yk )k∈N be such that lim yk = 0, yk > 0 and | f (yk ) − f (0)| > ε for k ≥ 1. k→∞

Let (xn )n∈N be a subsequence of (yk )k∈N such that xn+1 < xn (1 − c) for n ≥ 1. (1−c) 2xn Let an = 2xn2−c and bn = 2−c for n ≥ 1. Then bn+1 < an < bn , bn + an = 4xn −2cxn bn −an 2cxn = 2xn and bn = 2xn = c for all n ≥ 1. Define g : R → R by 2−c

8. Path continuity connected with density and porosity

g(x) =

  0

x ∈ {0} ∪

∞ S

123

[an , bn ],

n=1

∞ S   f (0) − f (x) + ε, x ∈ (−∞, 0) ∪ (bn+1 , an ) ∪ (b1 , ∞). n=1

  ∞ S Obviously, R \ {0} ⊂ Pr (g). Since p+ R \ [an , bn ], 0 = c > r, 0 ∈ Pr (g). n=1

Thus g ∈ Pr . Let E = {x : | f (x) + g(x) − f (0) − g(0)| < ε}. Certainly, f (0) + g(0) = f (0) and | f (x) + g(x) − f (0)| = ε for x ∈ (−∞, 0) ∪

∞ S

(bn+1 , an ) ∪

n=1

(b1 , ∞). Therefore 1 (bn − an ) xn − an = lim sup 2 1 = xn n→∞ n→∞ bn − 2 (bn − an ) 1 1 c = lim sup 2 = 2 = < r. 2−c n→∞ bn −an − 1 c −1

p(R \ E, 0) = p+ (R \ E, 0) = lim sup

bn

Hence f + g ∈ / Pr and f ∈ / Ma (Pr ). It follows that Ma (Pr ) ⊂ C . Inclusions Ma (Mr ) ⊂ C and Ma (Sr ) ⊂ C can be proved similarly.

t u

Remark 8.3. Maximal additive classes for S0 , P0 , M1 and for v-porouscontinuous functions are still unknown. Definition 8.16. Let F be a family of real functions defined on R. A set Mm (F ) = {g : R → R : ∀ f ∈F f · g ∈ F } is called the maximal multiplicative class for F . Remark 8.4. Let f : R → R, f (x) = 1 for x ∈ R be a constant function. If f ∈ F then Mm (F ) ⊂ F , Definition 8.17. [14] Let 0 < ρ < 1 and let Z(ρ) be the family of all measurable functions f : R → R such that at each x0 ∈ Dap ( f ) the following two conditions hold (Z1) f (x0 ) = 0 (in other words Dap ( f ) ⊂ N f ), (Z2) for each measurable set E such that E ⊃ N f and d(E, x0 ) > ρ we have lim d(E ∩ {x : | f (x)| < ε}, x0 ) > ρ.

ε→0+

Corollary 8.9. The family of approximately continuous functions is a proper subset of Z(ρ).  Theorem 8.33. [14] Mm U C ρ = Z(ρ) for each 0 < ρ < 1.

124

Stanisław Kowalczyk and Małgorzata Turowska

Definition 8.18. [14] Let Z(1) be the family of all measurable functions f : R → R such that at each x0 , at which f is not T ∗ -continuous, the following two conditions hold: (Z3) f (x0 ) = 0, (Z4) for each measurable set F such that d(F, x0 ) = 1 and N f ⊂ F and for each ε > 0 we have  d F ∩ {x : | f (x)| < ε}, x0 = 1. Corollary 8.10. The class of T ∗ -continuous functions is a proper subset of Z(1). Theorem 8.34. [14] Mm (U C 1 ) = Z(1). Definition 8.19. [16] Let 0 < λ ≤ ρ < 1. Let P(λ , ρ) be a set of all functions f : R → R satisfying the following conditions (P1) Dap ( f ) ⊂ N f , (P2) for each x ∈ Dap ( f ) and for each measurable set E such that E ⊃ N f and d(E, x) > λ , d(E, x) > ρ we have lim d(E ∩ {y : | f (y) − f (x)| < ε}, x) > λ

ε→0+

and lim d(E ∩ {y : | f (y) − f (x)| < ε}, x) > ρ.

ε→0+

Corollary 8.11. Let 0 < λ ≤ ρ < 1. Then A $ P(λ , ρ). Theorem 8.35. [16] Mm (C[λ ,ρ] ) = P(λ , ρ) for each 0 < λ ≤ ρ < 1. Definition 8.20. [17] For each ρ ∈ (0, 1) let A(ρ) be the family of all functions from OU C ρ such that at each x0 ∈ D( f ) the following two conditions hold (Aρ 1) f (x0 ) = 0, (Aρ 2) for every ε > 0 there exists x ∈ (x0 − ε, x0 + ε) such that λ (N f ∩ J) > ρλ (J), where J is a closed interval with endpoints x and x0 (in other words x0 is a point of ρ-type upper density in O’Malley sense of N f ). Definition 8.21. [17] For each ρ ∈ (0, 1) let B(ρ) be a family of all functions from OU C ρ such that at each x0 ∈ D( f ) the following two conditions hold (Bρ 1) f (x0 ) = 0, (Bρ 2) for every ε > 0 there exists x ∈ (x0 − ε, x0 + ε) such that λ (N f ∩ J) ≥ ρλ (J), where J is a closed interval with endpoints x and x0 .

8. Path continuity connected with density and porosity

125

Theorem 8.36. [17] 1. A(ρ) ⊂ Mm (OU C ρ ) for ρ ∈ (0, 1). 2. Mm (OU C ρ ) ⊂ B(ρ) for ρ ∈ (0, 1). Problem 8.1. Characterize Mm (OU C ρ ) for ρ ∈ (0, 1). Remark 8.5. Maximal multiplicative classes for porouscontinuous functions and v-porouscontinuous functions are still unknown.

References [1] K. Banaszewski, Funkcje ciagłe ˛ wzgl˛edem systemu s´cie˙zek, Doctoral Thesis, Łód´z, 1995. [2] K. Banaszewski, On ε-continuous functions, Real Anal. Exchange 21(1) (1995/96), 203-215. [3] K. Banaszewski, Algebraic properties of ε-continuous functions, Real Anal. Exchange 18 (1992/93), 253-268. [4] J. Borsík, Some classes of strongly quasicontinuous functions, Real Anal. Exchange 30, (2004/05), 689-702. [5] J. Borsík, J. Holos, Some properties of porouscontinuous functions, Math. Slovaca 64 (2014), No. 3, 741-750. [6] A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Mathematics, Vol. 659, Springer-Verlag Berlin Heidelberg New York, 1978. [7] A. M. Bruckner, R. J. O’Malley, B. S. Thomson, Path Derivatives: A Unified View of Certain Generalized Derivatives, Trans. Amer. Math. Soc. 283 (1984), 97-125. [8] A. Denjoy, Sur les fonctions dérivées sommables, Bull. Soc. Math. France, 43 (1915), 161-248. [9] E. P. Dolženko, Boundary properties of arbitrary functions, Math. USSR Izv. 31 (1967), 3-14 (in Russian). [10] T. Filipczak, On some abstract density topologies, Real Anal. Exchange 14 (1988/89), 140-166. [11] A. Karasi´nska, E. Wagner-Bojakowska, Some remarks on ρ-upper density, Tatra Mt. Math. Publ. 46 (2010), 85-89. [12] S. Kowalczyk, On preponderantly continuous functions, Pr. Nauk. Akad. Jana Długosza Cz˛est. Mat. XIV (2009), 75-86. [13] S. Kowalczyk, K. Nowakowska, A note on ρ-upper continuous functions, Tatra Mt. Math. Publ. 44 (2009), 153-158. [14] S. Kowalczyk, K. Nowakowska, Maximal classes for ρ-upper continuous functions, J. Appl. Anal. 19 (2013), 69-89. [15] S. Kowalczyk, K. Nowakowska, A note on the [0]-lower continuous functions, Tatra Mt. Math. Publ. 58 (2014), 111-128. [16] S. Kowalczyk, K. Nowakowska, Maximal classes for the family of [λ , ρ]-continuous functions, Real Anal. Exchange 36 (2010/11), 307-324.

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[17] S. Kowalczyk, K. Nowakowska, On O’Malley ρ-upper continuous functions, Math. Slovaca (in printing). [18] S. Kowalczyk, M. Turowska, Structural property of porouscontinuous functions, (submitting). [19] S. Kowalczyk, M. Turowska, Some generalizations of porosity and porouscontinuity, (submitting). [20] M. Marciniak, R. Pawlak, On the restrictions of functions. Finitely continuous functions and path continuity, Tatra Mt. Math. Publ. 24 (2002), 65-77. [21] M. Marciniak, On path continuity, Real Anal. Exchange 29(1) (2003/2004), 247-255. [22] J. Masterson, A nonstandard result about path continuity, Acta Math. Hungar. 59, Issue 1-2, 1992, 147-149. [23] K. Nowakowska, On a family of [λ , ρ]-continuous functions, Tatra Mt. Math. Publ. 44 (2009), 129-138. [24] K. Nowakowska, M. Turowska, A note on weakly ρ-upper continuous functions, Pr. Nauk. Akad. Jana Długosza Cz˛est. Mat. XIX (2014), 195-206. [25] R. J. O’Malley, Note about preponderantly continuous functions, Revue Roumanie Math. Pureed Appl. 21 (1976), 335-336. [26] D. N. Sarkhel, A. K. De, The proximally continuous integrals, J. Aust. Math. Soc. (series A) 31 (1981), 26-45. [27] B. Thompson, Real functions, Lecture Notes in Mathematics, vol. 1170, SpringerVerlag Berlin Heidelberg New York, 1985. [28] W. Wilczy´nski, Density topologies, Handbook of measure theory, chapter 15, Elsevier (2012), 307-324. [29] L. Zajíˇcek, Porosity and σ -porosity, Real Anal. Exchange 13 (1987/88), 314-350.

S TANISŁAW KOWALCZYK Institute of Mathematics, Pomeranian University in Słupsk ul. Arciszewskiego 22d, 76-200 Słupsk, Poland E-mail: skowalczyk@apsl.edu.pl

M AŁGORZATA T UROWSKA Institute of Mathematics, Pomeranian University in Słupsk ul. Arciszewskiego 22d, 76-200 Słupsk, Poland E-mail: malgorzata.turowska@apsl.edu.pl

Chapter 9

Topological similarity of functions

IVAN KUPKA

2010 Mathematics Subject Classification: 54C99, 26A15, 54C05, 90C99. Key words and phrases: topological behavior, continuity preserving, generalized continuity, function spaces, optimization.

9.1 Introduction In this text we will work with several topological notions that enable us to compare the behavior of functions. So-called relations of continuity and relations of constancy were introduced for the first time in [8]. We show on examples that these relations between functions occur naturally. This approach allows not only to compare functions, but also to generalize some "continuitypreserving" theorems and to generate new ones. Sometimes it allows to replace differentiation by a simpler procedure - manipulation with inequalities - that can be used to examine nondifferentiable functions too. As the reader will see later, comparing functions in this way yields also new insight into the structure of some function spaces. Optimization applications of this new approach will be shown as well. The most results presented here come from our articles [8], [9], we have slightly improved some of them. A few results are new (e.g. Lemma 9.1). If we use results of other authors, we always cite their works.

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9.2 Motivation New notions mentioned above are defined in the next section. Before defining them, we want the reader to know that these notions did not come just out of blue. Therefore we think it is important to show some preliminary examples of behavior of functions, known by everybody and relevant to our case. Let us consider the following simple situation. We have two continuous real functions of real variable - f and g - and we are counting a limit of their quotient applying the L‘Hospital‘s rule. Suppose a ∈ R, f and g have a finite derivative on open intervals (a − ε, a) and (a, a + ε) and lim f (x) = lim g(x) = x→a x→a 0 is true. Suppose we obtain 0

f (t) f (x) = lim 0 = 3. t→a g (t) x→a g(x) lim

The equalities above imply that there exists an interval I = (a − δ , a + δ ) 0

(t) such that 1 < gf 0 (t) < 4 for all t from I \ {a}. Now we can observe that f and g behave "similarly" on the interval I. Indeed, since none of the derivatives equals zero for a t from I \ {a}, they are both positive or both negative on (a − δ , a) (and on (a, a + δ )). There are four possibilities for f and g:

• f and g are increasing on I; • f and g are decreasing on I; • f and g are both increasing on (a − δ , a) and decreasing on (a, a + δ ) so they have both a local maximum at a; • f and g are both decreasing on (a − δ , a) and increasing on (a, a + δ ) so they have both a local minimum at a. Could we obtain an information like this without using derivatives? Well, we could argue, that f and g behave similarly because we know that for every b from I \ {a} the following is true f (b)− f (a) g(t2 ) is true. We are going to work with (i), (ii) can be reduced to (i) by working with the function −g, because −g ≈ f holds too. Supposing (i) is true, denote I = g([t1 ,t2 ]), J = g([t2 ,t3 ]). Of course I and J are closed intervals and [max{g(t1 ), g(t3 )}, g(t2 )] ⊂ I ∩ J. Pick an arbitrary point c from [max{g(t1 ), g(t3 )}, g(t2 )]. We can see that there exists two points o1 ∈ (t1 ,t2 ), o2 ∈ (t2 ,t3 ) such that c = g(o1 ) = g(o2 ). Since g ≈ f we obtain f (o1 ) = f (o2 ) and since g(t2 ) 6= g(o1 ), we have f (t2 ) 6= f (o1 ) too. Now remembering that o1 < t2 < o2 we see that f is not monotonous on [c, d]. (2) If f is strictly monotonous on [c, d] then it is monotonous on this interval and according to (1) g is monotonous too. Now it suffices to show that g is injective on [c, d]. But this has to be true because f ≈ g is true and f is injective on [c, d]. Before proving (3) and (4) we should realize that only the case of global extremum on an interval needs to be treated. This is so because a local extremum on an interval is a global extremum on a subinterval. (3) Suppose x from (a, b) is a point of a global extremum of f . Without loss of generality we are going to assume that f has a global maximum at x. Notice that since f ≈ g the sets f −1 ( f (x)) and g−1 (g(x)) are identical. If f is constant on [a, b], then g is constant on [a, b] too and we are done. Now we examine the second case - the case when the set g−1 (g(x)) does not coincide with ]a, b]. Choose a point t from [a, b] such that g(t) 6= g(x). Suppose g(t) > g(x) (the case g(t) < g(x) is similar and therefore omitted). We will show that for all z from < a, b > we have g(z) ≥ g(x). Suppose this is not true. Then there exists a point s from [a, b] such that g(s) < g(x). Suppose t < x < s ( other cases, for example t < s < x etc can be treated with the same reasoning that we use for our chosen case). Since the sets g−1 (g(t)) , g−1 (g(s)) and g−1 (g(x)) are pairwise disjoint, the sets f −1 ( f (t)), f −1 ( f (s)) and f −1 ( f (x)) are pairwise disjoint too. Examine the case f (s) < f (t) < f (x) (the case f (t)
⊂ f ([x, s]). Because of the definition of c and the continuity of f we obtain f (c) = f (t). This means g(c) = g(t). Since g is continuous the set g([c, s]) contains the closed interval [g(s), g(c)]. Since g(s) < g(x) < g(t) = g(c) is true, there exists a point r from the open interval (c, s) such that g(r) = g(x). This implies f (r) = f (x). But r is from (c, s) and because of the definition of c the point f (x) = f (r) is not from f ([c, s]). This is a contradiction. We have just proved that for all z from [a, b] the inequality g(z) ≥ g(x) holds. The function g is proven to have a global extremum at x. (4) Suppose f has a strict global maximum at x from (a, b). The (3) proven, we can claim that g has a global extremum at x. If this extremum of g would not be strict, there would exist a point c from [a, b] with the property g(c) = g(x). Since f ≈ g this would imply f (c) = f (x), but this is not possible. t u Now we are ready for the main result of this section. Theorem 9.7. Let X be a topological space, let x be from X. Let X be locally arcwise connected at x. Let f : X → R, g : X → R be continuous functions. Let f ≈ g. Then (j) x is a point of a local extremum of f if and only if x is a point of a local extremum of g (jj) x is a point of a strict local extremum of f if and only if x is a point of a strict local extremum of g Proof. If x is an isolated point the theorem is true. Suppose x is not isolated. (j) We will prove that if f has a local extremum at x then g has an extremum at x too. Suppose f has at x a local maximum. This means there exists an arcwise connected open neighborhood U of x such that for all t from U the inequality f (t) ≤ f (x) takes place. Choose an arbitrary point u from U \ {x}. Suppose g(u) ≥ g(x). We are going to prove that for all s from U the inequality g(s) ≥ g(x). Choose an arbitrary s from U, suppose s is different from x and u. Since U is arcwise connected, there exists an arc connecting the points u, x and s. More concretely there exists a continuous function h : [0, 2] → U such that h(0) = u, h(1) = x and h(2) = s. Define functions f : [0, 2] → R and g : [0, 2] → R in the following way for all z from [0, 2], f (z) = f (h(z)) and g(z) = g(h(z)). According to Lemma 9.4 the functions f and g satisfy f ≈ g. Since f has a local extremum on U at the point x, we can see that f has a local extremum on [0, 2] at the point 1. This means (according to Lemma 9.6) that g has a local

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extremum on [0, 2] at the point 1. We know that g(0) = g(u) ≤ g(x) = g(1) so g has at 1 a local minimum. Therefore g(2) ≥ g(1) must be true. Since g(2) = g(s) and g(1) = g(x) we have just proved that for an arbitrary s from U we have g(s) ≥ g(x). The next example shows that without connectedness of X our theorem need not be true. Example 9.2. Define a subset X of R by       X = −1, − 21 ∪ − 14 , − 81 ∪ · · · ∪ {0} ∪ 12 , 1 ∪ . . . More concretely, X = {0} ∪ A ∪ B where   ∞  ∞ ∞  ∞ [ [ [ [ 1 1 1 1 − 2i , − 2i+1 = Ii , B = A= , = Ji . 2 2 22i+1 22i i=0 i=0 i=0 i=0 We define two functions f : X → R and g : X → R by  1  1 x ∈ Ii ,   − 22i , x ∈ Ii , − 22i , g(x) = 0, f (x) = 0, x = 0, x = 0,    1  1 − 22i+1 , x ∈ Ji , , x ∈ Ji . 22i+1 The functions f and g coincide on A ∪ {0}. Globally, it is easy to see that f ≈ g is true. The function f has a strict global maximum at 0, but g is nondecreasing on its domain and has no extremum at 0. The following theorem will enable us to prove easily two optimization theorems. Theorem 9.8. Let X be a nonempty set, Y and Z be Hausdorff topological spaces. Let f : X → Y , g : X → Z be functions. Let f ≈ g. Then for any subset A of X the following is true: f (A) is closed (compact) in Y if and only if g(A) is closed (compact) in Z. Proof. First we will prove the "closedness" part of our assertion. It suffices to show that if f (A) is closed in Y then g(A) is closed in Z. Let {zγ }γ∈Γ be a net of points from g(A), which is convergent in Z. Denote its limit by z. We have to prove that there exists a point a in A such that g(a) = z is true. Since each point zγ is from g(A), for every γ from Γ there exists xγ from A such that g(xγ ) = zγ . We see that {g(xγ )}γ∈Γ = {zγ }γ∈Γ converges in Z. Together with f ≈ g this implies the net { f (xγ )}γ∈Γ converges in Y . We will denote its limit by y. The

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set f (A) is closed, so y ∈ f (A). Consider a point a ∈ A such that f (a) = y. Now let us consider the net {pγ ‘ }γ ‘ ∈Γ ‘ := {xγ , a}. We can see that lim f (pγ ‘ ) = γ ‘ ∈Γ ‘

y = f (a). Since f ≈ g this means there exists m ∈ Z such that m = lim g(pγ ‘ ). γ ‘ ∈Γ ‘

But the nets {xγ }γ∈Γ and {a}γ∈Γ (by this we mean the net {aγ }γ∈Γ where for all γ from Γ aγ = a) are both subnets of the net {pγ ‘ }γ ‘ ∈Γ ‘ . This implies: lim g(pγ ‘ ) = lim g(xγ ) = z and lim g(pγ ‘ ) = lim g(a) = g(a). So z = g(a) and γ ‘ ∈Γ ‘

γ∈Γ

γ ‘ ∈Γ ‘

γ∈Γ

this means also z ∈ g(A). The closedness of g(A) is proven. Now, for "compactness" part of out assertion, again, it suffices to prove, that if f (A) is compact, then g(A) is compact too. Suppose f (A) to be compact. Then it is closed so g(A) is closed too. Consider an arbitrary net {g(xγ )}γ∈Γ in g(A), we have to prove now, that it has a convergent subnet. But the net { f (xγ )}γ∈Γ in f (A) has a convergent subnet, say { f (xδ )}δ ∈∆ and because of strong similarity of f and g the net {g(xδ )}δ ∈∆ is convergent (in g(A)) too. t u Now, the following theorems will be easy corollaries of the preceding theorem. Let us observe that we do not need any topological structure on the domain set X in the following theorem. Theorem 9.9. Let X be a set. Let f : X → R, g : X → R be functions. Let A be a subset of X such that the set f (A) is closed. Let f achieve its maximum and minimum on A. Let f ≈ g. Then g achieves its maximum and minimum on A too. Proof. Under the conditions of our theorem the set f (A) must be compact. So the set g(A) is a compact too. t u Now we present an optimization result concerning functions, that are strongly similar with Darboux functions. Theorem 9.10. Let X be a topological space. Let f : X → R, g : X → R be functions. Let f be a Darboux function (an image of each connected set under f is connected). Let f achieve its maximum and minimum on a connected subset A of X. Let f ≈ g. Then g achieves its maximum and minimum on A too. Proof. Under the conditions of our theorem the set f (A) must be connected and bounded. Moreover it contains its supremum and infimum. So f (A) is a compact interval. This means the set g(A) is compact too. Therefore g achieves a global maximum and a global minimum on A. t u

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To conclude this section, let us remark, that all results presented here showed some possibilies how to investigate a nondifferentiable function for extrema. Namely, some nondifferentiable functions are strongly similar to differentiable ones and these can be investigated in a classical way. For the same reason it is also worth investigating which continuous functions defined on convex sets are strongly similar to convex functions.

References [1] D. P. Bertsekas, Nonlinear Programming, 2nd Ed., Athena Scientific, Belmont 1999. [2] J. Borsík, Bilateral quasicontinuity in topological spaces, Tatra Mt. Math. Publ. 28 (2004), 159-168. [3] J. Borsík, On the points of bilateral quasicontinuity of functions, Real Anal. Exchange 19 (1993/1994), 529-536. [4] J. Borsík, L’. Holá, D. Holý, Baire spaces and quasicontinuous mappings, Filomat 25 (2011), 69-83. [5] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg 1985. [6] R. Engelking, General Topology, PWN Warszaw 1977. [7] R. V. Fuller, Relations among continuous and various noncontinuous functions, Pac. J. Math. 25(3) (1968), 495-509. [8] I. Kupka, On similarity of functions, Top. Proc. 36 (2010), 173-187. [9] I. Kupka, Similar functions and their properties, Tatra Mt. Math. Publ. 55 (2013), 47-56. [10] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41. [11] M. Matejdes, On selections of multifunctions, Math. Bohem. 118 (1993), 255-260. [12] M. Matejdes, Quasicontinuous selections of upper Baire continuous mappings, Mat. Vesnik 62 (2010), 69-76. [13] M. Matejdes, Selection theorems and minimal mappings in a cluster setting, Rocky Mountain J. Math. 41 (2011), no. 3, 851-867. [14] T. Neubrunn, Quasicontinuity, Real Anal. Exchange 14 (1988-89), 259-306. [15] O. Njåstad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961-970. [16] B. Novotný, On subcontinuity, Real Analysis Exchange 31 (2) (2005/2006), 535-546. [17] Z. Piotrowski, A survey of results concerning generalized continuity on topological spaces, Acta Math. Univ. Comenian. 52-53 (1987), 91-110. [18] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1986.

I VAN K UPKA Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics Comenius University, Mlynska Dolina, Bratislava, Slovakia E-mail: kupka@fmph.uniba.sk

Chapter 10

On the Darboux property of derivative multifunction

˙ ´ GRAZYNA KWIECINSKA

2010 Mathematics Subject Classification: 26E35, 54C60, 54C05, 58C25, 46G05. Key words and phrases: multifunctions, Darboux property, derivative multifunctions.

10.1 Introduction The concept of differentiability for multifunction has been considered by many authors from different point of view ([1], [2], [3], [5], [6], [10], [11], [12], [13]). We need to differentiate multifunctions as much as we need differentiate single-valued maps, for extending Darboux theorem on intermediate value property of derivative to multifunctions for instans, and for many other reasons. How should we go about it? It is possible to define derivatives as adequate limits of differential quotients ([10], [11]). Starting from such a definition of derivative and using a theory of some new derivatives of single-valued functions given by Garg in [8] we will show that derivative multifunction has the Darboux property.

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10.2 Preliminaries We will use standard notations. In particular, the sets of positive integers and real numbers will be denoted by N and R, respectively. Rn will denote the n-dimensional Euclidean space. Let X and Y be two nonempty sets. By a multifunction from X to Y we mean a map which assigns to every point of X a nonempty subsets of Y ; if F is a multifunction from X to Y , we denote it by F : X Y . The image of a set A ⊂ X under multifunction F : X Y is defined by F(A) =

[

{F(x) : x ∈ A}.

If F : X Y is amultifunction, then for a set B ⊂ Y two inverse images of B under F are defined as follows: F + (B) = {x ∈ X : F(x) ⊂ B}, F − (B) = {x ∈ X : F(x) ∩ B 6= 0}. / One sees immediately that F − (B) = X \ F + (Y \ B) and

F + (B) = X \ F − (Y \ B).

Let (X, T (X)) be a topological spaces. We will use the notations IntA, ClA and FrA for the interior, closure and boundary of A ⊂ X, respectively. Let us still establish that (Y, T (Y )) is also a topological space. A multifunction F : X Y is called upper (resp. lower) semicontinuous at a point x ∈ X if (1) ∀G ∈ T (Y )(F(x) ⊂ G ⇒ x ∈ IntF + (G)) (resp. ∀G ∈ T (Y )(F(x) ∩ G 6= 0/ ⇒ x ∈ IntF − (G))). F is called continuous at x if it is simultaneously upper and lower semicontinuous at x; F is continuous if it is continuous at each point x ∈ X. Now suppose that (Y, d) is a metric space. Let y ∈ Y and A ⊂ Y . We use S B(y, r) to denote an open ball in Y and B(A, r) = {B(y, r) : y ∈ A}. In this case we have a set of more adjectives. A multifunction F : X Y is called h-upper (resp. h-lower) semicontinuous at a point x0 ∈ X if the following condition holds: (2) for each ε > 0 there exists a neighbourhood U(x0 ) of x0 such that F(x) ⊂ B(F(x0 ), ε) (resp. F(x0 ) ⊂ B(F(x), ε)) for each x ∈ U(x0 ).

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F is h-continuous (or Hausdorff continuous) at x0 if it is simultaneously hupper and h-lower semicontinuous at x0 ; F is h-continuous if it is h-continious at any point x ∈ X. It is known (see [9]) that (3) If F is upper (resp. h-lower) semicontinuous at x ∈ X, then F is h-upper (resp. lower) semicontinuous at x. If moreover F(x) is compact for each x ∈ X, then conditions (1) and (2) are equivalent. Let P0 (Y ) denote the family of all nonempty subsets of Y . We denote the following families of sets: C (Y ) = {A ∈ P0 (Y ) : A is closed} Cb (Y ) = {A ∈ P0 (Y ) : A is closed and bounded}. For A, B ∈ Cb (Y ) let dH (A, B) denotes the Hausdorff distance of the sets A and B. Then the set Cb (Y ) with Hausdorff distance becomes a metric space. Let us note that (4) If F : X Y has closed and bounded values, then F is h-continuous if and only if F is continuous (with respect to dH ) as a function from X to Cb (Y ). Let (Y, k · k) be a real normed linear space. The symbol Cob (Y ) will be used to denote the collection of all nonempty, closed, bounded and convex subsets of Y . If A ⊂ Y , B ⊂ Y , λ ∈ R and α ∈ R \ {0}, one defines A + B = {a + b : a ∈ A, b ∈ B}, λ A = {λ a : a ∈ A}; A − B = A + (−1)B

and

1 A = A. α α

We will write A + x, if B = {x}. (5) The following properties hold (see [14]): (i) If α, β ∈ R and A, B ⊂ Y are convex, then α(β A) = (αβ )A and α(A + B) = αA + αB. (ii) If A ⊂ B and α ≥ 0, then αA ⊂ αB. (iii) If A is convex, α ≥ 0 and β ≥ 0, then (α + β )A = αA + β A. (iv) If A ⊂ Y and B ⊂ Y are closed and convex and C ⊂ Y is bounded, then A +C = B +C implies A = B. (v) If (Y, k · k) is reflexive and A, B ∈ Cob (Y ), then A + B ∈ Cob (Y ).

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10.3 The D and D∗ properties of multifunctions As we know a real function f defined on an interval I ⊂ R has the Darboux property if for each pair of distinct points x1 , x2 ∈ I and each y between f (x1 ) and f (x2 ) there is a point x3 between x1 and x2 such that y = f (x3 ). It is well known that f has the Darboux property if and only if f (C) is a connected set for each connected set C ⊂ I. It turns out that if we extend given above properties into multifunctions they are not already equivalent. Let I ⊂ R be an interval. For each a, b ∈ R we will use a ∧ b and a ∨ b to denote the minimum and maximum, respectively, of a and b. In [7] the following definition of the Darboux property was given. Definition 10.1. A multifunction F : I R will be said to have the Darboux property (or D property) if for every connected set C ⊂ I, the image F(C) is connected in R. In [4] the following definition was introdused. Definition 10.2. A multifunction F : I R will be said to have the intermediate value property (or D∗ property) if for each pair of distinct points x1 , x2 ∈ I and each y1 ∈ F(x1 ) there exists y2 ∈ F(x2 ) such that (y1 ∧ y2 , y1 ∨ y2 ) ⊂ F((x1 ∧ x2 , x1 ∨ x2 )). Let us note that each of the properties D and D∗ is equivalent to the usual Darboux property in the case when F(x) = { f (x)}, where f : I → R is a function. The following examples show that they are not equivalent in general. Example 10.1. Let F1 : R

R be a multifunction defined by  [0, 2], if x = 0, F1 (x) = [0, 1], if x 6= 0.

Then F1 has the D property , but not the D∗ property. Example 10.2. Let F2 (x) = [0, 1] ∪ [2, 3] for each x ∈ R. Then F2 has the D∗ property and does not have the D property . However they showed the following theorem. Theorem 10.1. ([4], Theorem 1]) Let F : I R be a multifunction with connected values. If F has the D∗ property, then it has the D property.

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Let us note that F2 is continuous. Therefore, a continuous multifunction (with closed values) does not necessarily have the D property unlike the case of the D∗ property. The following theorem was proved. Theorem 10.2. ([4], Theorem 2]) If a multifunction F : I ues is continuous, then it has the D∗ property.

R with closed val-

Remark 10.1. The assumption that a multifunction has closed values is important. In order to illustrate this, let us consider a multifunction F : R R defined by  if x ∈ (0, 1), {y : y = 1k , k ∈ Z \ {0}}, F(x) = {y : y = 0 or y = 1k , k ∈ Z \ {0}}, if x ∈ / (0, 1), where Z is the set of integers. Then F is continuous but does not have the D∗ property .

10.4 Derivative multifunction Let (Y, k · k) be a reflexive real normed linear space with the metric d determined by the norm in Y ; θ will denote the neutral element of Y . We define a difference A B of the sets A, B ∈ Cob (Y ) as follows: Definition 10.3. We will say the difference A B is defined if there exists a set C ∈ Cob (Y ) such that either A = B +C or B = A −C, and we define A B to be the set C. Using property (5) (iv) it is easy to show, that the difference A B is uniquely determined. Example 10.3. Let A = αP and B = β P, where P ∈ Cob (Y ), α ≥ 0 and β ≥ 0. Let us put C = (α − β )P. Then, by (5) (iii), B +C = A or A −C = B depending on whether α ≥ β or α < β . Therefore αP β P exists and is equal to (α − β )P. Example 10.4. Consider the following sets:  A = (x, y) ∈ R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x ,  B = (x, y) ∈ R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 21 (1 − x) . Then A B does not exist. Indeed, suppose that there exists C ∈ Cob (R2 ) such that A = B +C. Since (0, 1) ∈ A, there exist (a, b) ∈ B and (c, d) ∈ C such that

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(0, 1) = (a + c, b + d), where a ≥ 0. Then c = −a and d = 1 − b. On the other hand (0, 0) ∈ B. Therefore (0, 0) + (c, d) = (−a, 1 − b) ∈ A and −a ≥ 0. Hence a = 0. Since (c, d) = (0, 1 − b) ∈ C and (1, 0) ∈ B, we have (1, 0) + (0, 1 − b) ∈ A and b = 1. Therefore we have (a, b) = (0, 1) ∈ / B, which is a contradiction. Now let us suppose that there exists C ∈ Cob (R2 ) such that B = A − C. Let z ∈ C. We observe that for every x ∈ A, x − z ∈ A − C = B. Hence we have A − z ⊂ B, i.e., some translate of A is contained in B, which is of course not possible. Let A, B ∈ Cbc (Y ). We write B ⊂t A, if for each a ∈ FrA there is y ∈ Y such that a ∈ B + y ⊂ A. Theorem 10.3. Suppose A ∈ Cob (Y ) and B ∈ Cob (Y ). Then (a) A B exists and is equal to a set C ∈ Cob (Y ) such that A = B + C if and only if B ⊂t A. (b) A B exists and is equal to a set C ∈ Cob (Y ) such that B = A − C if and only if A ⊂t B. Proof. To prove (a), suppose the existence of C ∈ Cob (Y ) such that A = B +C. If a ∈ A (in particular a ∈ Fr(A)), then a ∈ B + C. Therefore exist b ∈ B and c ∈ C such that a = b + c. If z ∈ B, then z + c ∈ B + C = A. Consequently B + c ⊂ A. Moreover a = b + c ∈ B + c. This proves that for a there is y ∈ Y with a ∈ B + y ⊂ A. Now let us suppose that for each a ∈ FrA there exists y ∈ Y such that a ∈ B + y ⊂ A. Assume that C = {x : B + x ⊂ A}. Then C is closed and bounded. We will show that C is convex. Let c, c0 ∈ C. Then B + c ⊂ A and B + c0 ⊂ A. Let λ ∈ [0, 1]. From (5) (ii) and (iii) we obtain (6)

(1 − λ )(B + c) + λ (B + c0 ) ⊂ A.

Furthermore (7)

(1 − λ )(B + c) + λ (B + c0 ) = B + (1 − λ )c + λ c0 .

We conclude from (6) and (7) that B + (1 − λ )c + λ c0 ⊂ A, hence that z = (1 − λ )c + λ c0 ∈ C, and finally that C is convex. Since B +C ⊂ A, we need to prove that A ⊂ B +C. Let x ∈ A. Since A is convex there exist a, a0 ∈ FrA and λ ∈ [0, 1] such that x = (1 − λ )a + λ a0 . Then by hypothesis there exist y, y0 ∈ Y such that a ∈ B + y ⊂ A and a0 ∈ B + y0 ⊂ A. Thus there exist b, b0 ∈ B such that a = b + y and a0 = b0 + y0 and x = (1 − λ )a + λ a0 = b00 + (1 − λ )y + λ y0 , where b00 = (1 − λ )b + λ b0 . Thus x ∈ B + (1 − λ )y + λ y0 . Since y, y0 ∈ C and C is convex, u = (1 − λ )y + λ y0 ∈ C. Therefore x ∈ B + C, which finishes the proof of (a).

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To prove (b) we apply similar arguments, with {x : B + x ⊂ A} replaced by {x : A − x ⊂ B} in the second part of the proof. t u It is easy to see that (8) If B ∈ Cob (Y ) and y ∈ Y then (B + y) B = {y}. In particular A A = {θ }. (9) If A B exists, then dH (A, B) = kA Bk, where kCk = dH (C, {θ }) for C ⊂ Y. (10) If Y = R and A, B ∈ Cob (R), then A B exists and A B = [(a − b) ∧ (x − y), (a − b) ∨ (x − y)], where A = [a, x] and B = [b, y]. Now we can present a definition of derivative of a multifunction. Definition 10.4. A multifunction F : I Y with F(x) ∈ Cob (Y ) is said to be differentiable at a point x0 ∈ I if there exists a set DF(x0 ) ∈ Cob (Y ) such that the limit (with respect to the Hausdorff metric) lim

x→x0

F(x) F(x0 ) x − x0

exists and is equal to DF(x0 ). The set DF(x0 ) will be called the derivative of F at x0 . F will be called differentiable if it is differentiable at every point x ∈ I. Of course, implicit in the definition of DF(x0 ) is the existence of the differences F(x) F(x0 ). Example 10.5. A multifunction F : [0, 1]

R2 defined by the formula

F(α) = {(x, y) ∈ R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ α − αx} is not differentiable, since the required differences do not exist (see Example 10.4). Theorem 10.4. If a multifunction F : I Y with closed, bounded and convex values is differentiable at x0 ∈ I, then it is h-continuous at this point. Proof. Suppose F is differentiable at x0 . Then we can assume that the differences F(x) F(x0 ) exist for x in some neighbourhood of x0 . Let x 6= x0 . By the differentiability of F at x0 , there exists a set DF(x0 ) ∈ Cob (Y ) such that

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 (11)

lim dH

x→x0

 F(x) F(x0 ) , DF(x0 ) = 0. x − x0

Then (see (9))

0) dH (F(x), F(x0 )) = kF(x) F(x0 )k = F(x) F(x

|x − x0 | ≤ x−x0     0) ≤ dH F(x) F(x , DF(x0 ) + kDF(x0 )k |x − x0 |. x−x0

(12)

Since the set DF(x0 ) is bounded, (11) and (12) shows that dH (F(x), F(x0 )) converges to zero as x tends to x0 . Hence, F is h-continuous at x0 , by (4). t u Definition 10.5. A multifunction G : I Y will be called a derivative if there exists a differentiable multifunction F : I Y with G(x) = DF(x) for each x ∈ I. Example 10.6. Let S be the closed unit ball in Y , and consider a multifunction F : (0, 2π) R2 defined by F(x) = (2 + sin x)S. Then F is differentiable and DF(x) = (cos x)S for each x ∈ (0, 2π) and the multifunction G : (0, 2π) Y given by G(x) = (cos x)S is a derivative. Now we will deal with the case when Y = R. Let F : I tion with compact and convex values. Then (13)

Y be a multivfunc-

F(x) = [i(x), s(x)],

where i(x) = inf F(x) and s(x) = sup F(x) for x ∈ I. It should be noted that in this case F(x) F(x0 ) exists for x ∈ I and h i i(x)−i(x0 ) s(x)−s(x0 )  , ,  x−x x−x h 0 0  i   s(x)−s(x ) i(x)−i(x ) 0 0 F(x) F(x0 ) h x−x0 , x−x0 i , (14) = s(x)−s(x0 ) i(x)−i(x0 )  x − x0  x−x0 , x−x0 i ,   h    i(x)−i(x0 ) , s(x)−s(x0 ) , x−x0 x−x0

if δ F(x) ≥ δ F(x0 ), x > x0 , if δ F(x) ≥ δ F(x0 ), x < x0 , if δ F(x) ≤ δ F(x0 ), x > x0 , if δ F(x) ≤ δ F(x0 ), x < x0 ,

where δ A denotes diameter of A. It can be verified without difficulty that Theorem 10.5. If the functions i : I → R and s : I → R are differentiable at x0 ∈ I, then multifunction F given by (13) is differentiable at x0 and

10. On the Darboux property of derivative multifunction

DF(x0 ) =

155

( [i0 (x0 ), s0 (x0 )], if i0 (x0 ) ≤ s0 (x0 ), [s0 (x0 ), i0 (x0 )], if i0 (x0 ) > s0 (x0 ).

However, in general, differentiability of F does not imply differentiability of the functions i or s as the following example shows: ( [0, x], if x ≥ 0, F(x) = [x, 0], if x < 0. But in the case F is differentiable at x0 , both functions i and s are either simultaneously differentiable at x0 or simultaneously nondifferentiable at x0 . As a consequence of Theorem 10.4 we have the following property. (15) If a multifunction F : I R given by (13) is differentiable at a point x0 ∈ I, then F is h-continuous at x0 and consequently the functions i and s are continuous at x0 . Let us suppose that the multifunction F given by (13) is differentiable at a point x0 ∈ I. According to Definition 10.4, there is a set DF(x0 ) ∈ Cob (R) such that (16)

lim

x→x0

F(x) F(x0 ) x−x0

= DF(x0 ).

This condition can be reinterpreted in terms of Dini derivatives of the functions i and s.

10.5 A new notion of derivative of functions Garg in [8] has presented a unified theory of Dini derivatives and a theory of some new derivatives of functions. After the discovery of Weierstrass, it became well known that there are continuous functions that are not derivable at any point. The same holds in terms of various generalized derivatives that are known, e.g. the Dini, approximate and symmetric derivatives. Garg showed that in terms of new derivatives every continuous function is derivable at a set of points which has cardinality continuum in every interval, and the properties of f can in turn be investigated in terms of the values of its new derivative. Many of the known results in differentiation theory, like the mean value theorems and the Darboux property of derivative, are found to hold in terms of new derivatives without any derivability hypothesis. Let f : I → R be a function and x ∈ IntI. We will use f−0 (x) and f+0 (x) to denote the left-side and right-side derivatives of f at x, D− f (x), D− f (x), D+ f (x)

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and D+ f (x) to denote the left and right lower and upper Dini derivatives of f at x. Further, given x, y ∈ I, x 6= y, we will use Q f (x, y) to denote the following difference quotient of f on [x, y] or [y, x]: Q f (x, y) =

f (x) − f (y) . x−y

Following Garg we are accepting the following definitions. Definition 10.6. A number c ∈ R = [−∞, ∞] is called a lower (resp. upper) gradient of f at x ∈ IntI if D− f (x) ≤ c ≤ D+ f (x) (D+ f (x) ≤ c ≤ D− f (x)). For example, for the norm function f (x) = |x| each element of the interval [−1, 1] is a lower gradient of f at 0. The lower and upper derivatives of f are also defined in terms of its Dini derivatives. Definition 10.7. A function f : I → R is lower (resp. upper) derivable at a point x ∈ IntI, if D− f (x) ≤ D+ f (x) (resp. D+ f (x) ≤ D− f (x)), and then the interval [D− f (x), D+ f (x)] (resp. [D+ f (x), D− f (x)]) is called the lower (resp. upper) derivative of f at x and denoted by L f 0 (x) (resp. Uf’(x)). So L f 0 (x) = [D− f (x), D+ f (x)] (resp. U f 0 (x) = [D+ f (x), D− f (x)]. We call further f semi-derivable at x, if it is either lower or upper derivable at x, and then its lower or upper derivative at x is called the semi-derivative of f at x and denoted by S f 0 (x). When the lower, upper or semi-derivative of f at x is a singleton, f is said to be uniquely lower, upper or semi-derivable , respectively, at x, and then L f 0 (x), U f 0 (x) or S f 0 (x) are also used to denote its unique element. Further, when f has a finite lower, upper or semi-gradient at x, f is called lower, upper or semi-differentiable, respectively, at x; and when this gradient is further unique, f is called uniquely lower, upper or semi-differentiable, respectively, at x. Also, when f has a finite ordinary derivative at x, f is called simply differentiable at x. As the lower, upper and semi-derivatives are set-valued, and they are not defined in terms of a limit, the nature of results on these derivatives are quite different from the usual results. They include, however, most of the results on the ordinary derivative. We quote now these properties which will be essential for the proof of the main theorem our paper . Let f be a function on R and I = [x1 , x2 ] ⊂ R. Moreover ∆S ( f ) will denote the set of all points in I where f is semi-derivable. Theorem 10.6. [[8], Theorem 8.1.2] If f is continuous relative to I, then there is a point x ∈ (x1 , x2 ) such that f is semi-derivable at x and Q f (x1 , x2 ) ∈ S f 0 (x).

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157

Theorem 10.7. [[8], Theorem 10.4.1] The semi-derivatives of every continuous relative to I function f posses the Darboux property in the following sense: S for each connected set C ⊂ R the set {S f 0 (x) : x ∈ C ∩ ∆S ( f )} is connected in the set R. Finally we define some set valued medians of f which are associated with lower and upper derivatives of f . Definition 10.8. If D+ f (x) ≤ D− f (x), then the interval [D+ f (x), D− f (x)] will be called the lower median of f at x; and when D− f (x) ≤ D+ f (x), the interval [D− f (x), D+ f (x)] will be called the upper median of f at x. We will use M f (x) and M f (x) to denote the lower and upper median, respectively, of f at the point x. Now let us return to the multifunction F given by (13). Let us suppose that x0 ∈ IntI, F is differentiable at x0 and DF(x0 ) = [a, b], where a, b ∈ R and a ≤ b. Then (14) and (16) force a and b to be the only limit points of Qi(x, x0 ) and Qs(x, x0 ). If a = b, then the four Dini derivatives of i and s at x0 are equal, and hence the functions i and s are differentiable at x0 with i0 (x0 ) = s0 (x0 ). If a < b and the functions i and s are not differentiable at x0 , then they have a semi-derivative at x0 or a lower or upper median at x0 . We consider this in four basically different cases. Case (i): There exists h > 0 such that δ F(x) ≥ δ F(x0 ) for each point x ∈ (x0 , x0 + h) and δ F(x) ≤ δ F(x0 ) for each x ∈ (x0 − h, x0 ). Note that in this case (16) holds if and only if D+ i(x0 ) = D+ i(x0 ) = a, D+ s(x0 ) = D+ s(x0 ) = b, and D− i(x0 ) = D− i(x0 ) = a, D− s(x0 ) = D− s(x0 ) = b. Thus the functions i and s are differentiable at the point x0 and DF(x0 ) = [i0 (x0 ), s0 (x0 )]. Of course, Li0 (x0 ) = Ui0 (x0 ) = a and Us0 (x0 ) = Ls0 (x0 ) = b. Case (ii): There exists h > 0 such that δ F(x) ≥ δ F(x0 ) for each x ∈ (x0 , x0 + h) and δ F(x) ≥ δ F(x0 ) for each x ∈ (x0 − h, x0 ). In this case (16) holds if and only if D+ i(x0 ) = D+ i(x0 ) = a, D+ s(x0 ) = D+ s(x0 ) = b, and D− s(x0 ) = D− s(x0 ) = a, D− i(x0 ) = D− i(x0 ) = b. Thus the function i is upper derivable at x0 , the function s is lower derivable at x0 , and Ui0 (x0 ) = [a, b] = Ls0 (x0 ), and DF(x0 ) = [i0+ (x0 ), s0+ (x0 )] = [s0− (x0 ), i0− (x0 )]. Case (iii): There exists h > 0 such that δ F(x) ≥ δ F(x0 ) for each x ∈ (x0 , x0 + h) but for each h > 0 there exists x ∈ (x0 − h, x0 ) such that δ F(x) ≥ δ F(x0 ) and there exists x0 ∈ (x0 − h, x0 ) such that δ F(x0 ) < δ F(x0 ). In this case (16) holds if and only if D+ i(x0 ) = D+ i(x0 ) = a and D+ s(x0 ) = D+ s(x0 ) = b, D− s(x0 ) = a and D− s(x0 ) = b, D− i(x0 ) = a and D− i(x0 ) = b. Thus Ui0 (x0 ) = a, Ls0 (x0 ) = b, and DF(x0 ) = Mi(x0 ) = Ms(x0 ) = [i0+ (x0 ), s0+ (x0 )].

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Case (iv): For each h > 0 there exists x ∈ (x0 , x0 + h) such that δ F(x) ≥ δ F(x0 ) and there exists x0 ∈ (x0 , x0 + h) such that δ F(x0 ) < δ F(x0 ), and for each h > 0 there exists x ∈ (x0 − h, x0 ) such that δ F(x) ≥ δ F(x0 ) and there exists x0 ∈ (x0 − h, x0 ) such that δ F(x0 ) < δ F(x0 ). In this case (16) holds iff D+ i(x0 ) = a and D+ s(x0 ) = b, D+ s(x0 ) = a and D+ i(x0 ) = b, D− s(x0 ) = a and D− i(x0 ) = b, D− i(x0 ) = a and D− s(x0 ) = b. Thus neither the function i nor the function s is semi-derivable at x0 , and Mi(x0 ) = Mi(x0 ) = Ms(x0 ) = Ms(x0 ) = DF(x0 ).

10.6 The D∗ property of derivative multifunction Now we restrict our attention to the well-known result on ordinary derivative of functions, namely the intermediate value property of derivative. We will extend this result to the multivalued case. Theorem 10.8. Suppose F : I R is a multifunction with compact and convex values. If F is a derivative, then F has the intermediate value property. Proof. Assume the contrary. Then (17) there exist two distinct points x1 , x2 ∈ I, say x1 < x2 , and a point y1 ∈ F(x1 ) such that for any y ∈ F(x2 ) there exists a number α with α ∈ (y1 ∧ y, y1 ∨ y) \ F((x1 , x2 )). Obviously y1 ∈ / F(x2 ). Let y2 = inf F(x2 ). We have either y1 < y2 or y1 > y2 . Let us suppose that y1 < y2 and (18)

α ∈ (y1 , y2 ) \ F((x1 , x2 )).

On the other hand, by hypothesis, there is a differentiable multifunction Φ: I R such that F(x) = DΦ(x) for each x ∈ I. It follows from Theorem 10.4 that Φ is h-continuous. Assume Φ(x) = [i(x), s(x)] (see (13)). Then the functions i and s are continuous (see (15)). Let [ K = {Si0 (x) : x ∈ (x1 , x2 ) ∩ ∆S (i)} and L=

{Ss0 (x) : x ∈ (x1 , x2 ) ∩ ∆S (s)},

[

where ∆S (i) and ∆S (s) denote the sets of points at which the functions i and s, respectively, are semi-derivable. By Theorem 10.7, both sets K and L are connected.

10. On the Darboux property of derivative multifunction

159

Let us notice that (19) If x ∈ [x1 , x2 ] and z ∈ {D+ i(x), D+ i(x), D− i(x), D− i(x)}, then z is a limit point of K. (20) If x ∈ [x1 , x2 ] and z ∈ {D+ s(x), D+ s(x), D− s(x), D− s(x)}, then z is a limit point of L. Indeed, without loss of generality we can assume that z = D+ i(x). Thus there is a sequence (xn )n∈N which converges to x from the right such that (21)

lim Qi(x, xn ) = z.

n→∞

We conclude from Theorem 10.6 that for each n ∈ N exists yn ∈ (x, xn ) such that the function i is semiderivable at yn and (22)

Qi(x, xn ) ∈ Si0 (yn ) ⊂ K.

By (21) and (22) we have (19). Similarly we can schow (20). Suppose that F(x1 ) = [p, q] and F(x2 ) = [y2 , r]. Then according to (18) we have (23)

p ≤ y1 < α ≤ y2 ≤ r.

Let us suppose that p ∈ {D+ i(x1 ), D+ i(x1 )}. One of the points y2 or r belongs to the set {D− i(x2 ), D− i(x2 )}. Suppose y2 . Then according to (19) p and y2 are the limit points of K. The set K is connected. Therefore (p, y2 ) ⊂ K and, by (23), α ∈ K. Similarly if p ∈ {D+ s(x1 ), D+ s(x1 )}, then α ∈ L. Therefore (24)

α ∈ K ∪ L.

Let us note that K ∪ L ⊂ F((x1 , x2 )). So, by (24) α ∈ F((x1 , x2 )). But this contradicts (18). We obtain a similar conclusion when y1 > y2 . This completes the proof of Theorem 10.8. t u Observe that, by Theorem 10.1 and Theorem 10.8, we have the following Corollary. Corollary 10.1. If F : I R is a derivative multifunction with compact and connected values, then F has the Darboux property.

References [1] J. P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston-Basel-Berlin (1990).

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[2] H. T. Banks, M. Q. Jacobs, A differential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970), 246-272. [3] T. F. Bridgland, Trajectory integrals of set valued functions, Pacific J. Math. 33 (1970), 43-67. [4] J. Czarnowska, G. Kwieci´nska, On the Darboux property of multivalued functions, Demonstr. Math. 25 (1992), 192-199. [5] F. S. De Blasi, On the differentiability of multifunctions, Pacific J. Math. 66 (1976), 67-81. [6] G. Debreu, Integration of correspondences, in: Proc. Fifth Berkeley Sympos. Mat5h. Statist. and Probability, Vol. II, Part I (1996), 351-372. [7] J. Ewert, J. Lipi´nski, On the continuity of Darboux multifunctions, Real Anal. Exchange 13 (1987/88), 122-125. [8] K. M. Garg, Theory of Differentiation, Canadian Mathematical Society, series of Monographs and Advanced Texts, vol. 24, A Wiley-Interscience Publications, 1998. [9] S. Hu, N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Kluwer Academic Publishers, 1997. [10] M. Hukuhara, Integration des applications measurables dont la valuer est un compact convexe, Funkcialaj Ekvacioj 10 (1967), 205-223. [11] G. Kwieci´nska, Measurability of multifunctions of two variables, Dissertationes Math. 452 (2008), 1-67. [12] A. Lasota, A. Strauss, Assymptotic behavior for differential equations which cannot be locally linearized, J. Differential Equations 10 (1971), 152-172. [13] M. Martelli, A. Vignoli, On differentiability of multi-valued maps, Bull. Un. Mat. Ital. (4) 10 (1974), 701-712. [14] H. Radström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169.

˙ ´ G RA ZYNA K WIECI NSKA Institute of Mathematics, Pomeranian University in Słupsk ul. Arciszewskiego 22d, 76-200 Słupsk, Poland E-mail: grazyna.kwiecinska@apsl.edu.pl

Chapter 11

On extensions of quasi-continuous functions

OLEKSANDR V. MASLYUCHENKO AND VASYL V. NESTERENKO

2010 Mathematics Subject Classification: 54C20, 54C10, 54C30, 26A15. Key words and phrases: quasi-continuous function, extension of function, upper limit function, lower limit function, quasi-open set, quasi-closed set, quasi-clopen set.

11.1 Introduction The Tietze-Urysohn Extension Theorem (see [1, Theorem 2.1.8]) states that every continuous function defined on a closed subspace of a normal topological space has a continuous extension on the whole space. But we may have problems if we want extend functions with other properties. In this chapter we deal with the problem of extension of quasi-continuous functions. Definition 11.1. A function f : X → Y between topological spaces X and Y is called quasi-continuous if for any point x ∈ X, for any neighborhood U of x and for any neighborhood V of f (x) there exists a non-empty open set U1 ⊆ U, such that f (U1 ) ⊆ V . In general, this problem may be formulated in such a way. Problem 11.1. Describe all topological spaces X, Y and Z with Y ⊆ X, such that every quasi-continuous function g : Y → Z has a quasi-continuous extension f : X → Z.

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We also will consider the following specification of the previous problem. Definition 11.2. Let X and Z be topological spaces, Y ⊆ Z, f : X → Z, g : Y → Z. Define C( f ) ={x ∈ X : f is continuous at x}, e ={x ∈ Y : there exists z ∈ Z such that g(y) → z as Y 3 y → x}, C(g) e f ). e f ) = Y \ C( D( f ) =X \C( f ) and D( e ∩Y = C(g) and D(g) e ∩Y = D(g). So, we have that C(g) = Obviously, C(g) e Y \ D(g) = Y \ D(g). Problem 11.2. Describe all topological spaces X, Y and Z with Y ⊆ X, such that every quasi-continuous function g : Y → Z has a quasi-continuous extene sion f : X → Z with D( f ) = D(g). We know only one article [8] where quasi-continuous extensions was constructed. But there were extensions of bounded continuous functions defined on a subset of the segment [0;1] of the reals. Theorem 11.1 (Neugebauer, [8, Theorem 4]). Let X = [0; 1] be a segment of the real line, Y be a subspace of X and g : Y → R be a bounded continuous function. Then there exists an quasi-continuous function f : X → R such that f |Y = g and f is continuous at every point of Y . The following example shows, that there is an unbounded continuous function without any quasi-continuous extensions. Remark 11.1. Let g : (0; 1] → R be the function defined by g(y) = 1y . Then g has no quasi-continuous extension f : [0; 1] → R. Now we say about the contents of this chapter. In Section 11.2 we obtain the positive answer on Problem 11.2 in the case, where Y = X and g is a bounded real-valued quasi-continuous function. The case of compact-valued functions is investigated in Section 11.3. In Section 11.4 we obtain the positive answer on Problem 11.2 in the case, where Y is a closed Baire subspace of a hereditarily normal space X and Z = R. The rest of this chapter relates with Problem 11.1. So, in Sections 11.5 and 11.6 we discuss a question of the existence of, so called, the universal quasi-continuous extensions. In Sections 11.7 and 11.8 we obtain some auxiliary statements about quasi-clopen sets and quasi-clopen partitions. One of the mains results of this chapter we obtain in Section 11.9. There we give the positive answer on Problem 11.1 in the case, where X is hereditarily normal and Z

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is metrizable compact. In the final Section 11.10 we construct two counterexamples and give some open problems on the existence of quasi-continuous extensions.

11.2 Extension of real-valued quasi-continuous functions defined on a dense subspace In this section we start from some modifications of methods used in the proof of Theorem 11.1 which allows to extent a bounded quasi-continuous function defined on subspace of a hereditarily Baire hereditarily normal space. Firstly, let us introduce some notations and definitions. Definition 11.3. Let Y be a subspace of a topological space X and let g : Y → R be a bounded function. The upper and lower limit functions of g are said to be functions g∨ , g∧ : Y → R defined by the formulas g∨ (x) = lim sup g(y) = inf sup g(y) Y 3y→x

and

U∈Ux y∈U∩Y

g∧ (x) = lim inf g(y) = sup inf g(y) Y 3y→x

U∈Ux y∈U∩Y

for any x ∈ Y , where Ux is the system of all neighborhoods of x in X. The oscillation of g is said to be a function ω f : Y → R defined by ω f (x) = lim sup |g(y0 ) − g(y00 )| = inf Y 3y0 →x

sup

U∈Ux y0 ,y00 ∈U∩Y

|g(y0 ) − g(y00 )|.

Y 3y00 →x

e It is easy to see that ω f = g∨ − g∧ and D(g) = {x ∈ Y : ωg (x) > 0}. Proposition 11.1. Let Y be a dense subspace of a topological space X and let g : Y → R be a bounded quasi-continuous function. Then there exists a bounded quasi-continuous function f : X → R such that f |Y = g and D( f ) = e D(g). Proof. Define f : X → R by the formula  g(x), x ∈ Y ; f (x) = g∨ (x), x ∈ X \Y. Obviously, f |Y = g and f is bounded. Let us prove that f is quasi-continuous. Fix x0 ∈ X, ε > 0 and an open neighborhood U0 of x0 .

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Firstly, consider the case, where x0 ∈ Y . Since g is quasi-continuous at x0 , there exists a non-empty open set V ⊆ U0 such that |g(x) − g(x0 )| < ε for any x ∈ V ∩Y . But g(x0 ) = f (x0 ). So, f (x0 )−ε < g(x) < f (x0 )+ε for all x ∈ V ∩Y . Since Y = X, we have that f (x0 ) − ε < g(x) ≤ g∨ (x) ≤ f (x0 ) + ε on V . Thus, |g∨ (x) − f (x0 )| ≤ ε for x ∈ V and then f is quasi-continuous at x0 . Now we consider the case, where x0 ∈ X \ Y . Since f (x0 ) = g∨ (x0 ), there exists an open neighborhood U ⊆ U0 of x0 such that sup g(y) < f (x0 ) + ε2 . y∈U∩Y

But sup g(y) ≥ g∨ (x0 ) = f (x0 ) > f (x0 ) − ε2 .

y∈U∩Y

So, there is a point y0 ∈ U ∩Y with g(y0 ) > f (x0 ) − ε2 . Then | f (y0 ) − f (x0 )| = |g(y0 ) − f (x0 )| < ε2 . By the previous case, we have that f is quasi-continuous at y0 . Thus, there is a non-empty open subset V of U such that | f (x) − f (y0 )| < ε2 on V . Therefore, | f (x) − f (x0 )| ≤ | f (x) − f (y0 )| + | f (y0 ) − f (x0 )| < ε on V and then f is quasi-continuous at x0 . e Finally, to prove the equality D( f ) = D(g) it is sufficient to show that ω f = ∨ ∨ ωg . Since f ≤ g and g is upper semi-continuous, we have that f ∨ ≤ g∨ . On the other hand, equalities f |Y = g and Y = X imply that f ∨ ≥ g∨ . So, f ∨ = g∨ . Analogously, since f ≥ g∧ and f |Y = g, we conclude that f ∧ = g∧ . Therefore, ω f = f ∨ − f ∧ = g∨ − g∧ = ωg . t u

11.3 Cluster sets and extension of compact-valued quasi-continuous functions defined on a dense subspace Now we generalize the result of the previous section to the case of functions ranged in compact spaces. Definition 11.4. Let X and Z be topological spaces, x ∈ X, Y ⊆ X and g : Y → R. The cluster set of g at x is defined by the formula g(x) =

\

g(U ∩Y ),

U∈Ux

where Ux is the system of all neighborhoods of x in X. Set D = {x ∈ Y : g(x) 6= 0}. /

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The multifunction g : D ( Z is called the cluster multifunction of g. So, the domain dom g of this multifunction g is equals to the set D. It is not hard to show, that the graph Gr g of the cluster multifunction g is the closure of the graph Gr g of g in X × Z. So, Y ⊆ dom g ⊆ Y and in the case, where Z is compact we have, that dom g = Y . Definition 11.5. A multifunction F : X ( Y between topological spaces X and Y is called minimal if for any x0 ∈ X and any open sets U in X and V in Y , such that x0 ∈ U and V ∩ F(x0 ) 6= 0, / there is a nonempty open set U1 in U, such that F(U1 ) ⊆ V . Proposition 11.2. Let X be a topological spaces, Z be a regular space, Y ⊆ X and g : Y → Z be a quasi-continuous function. Then the cluster multifunction g is minimal. Proof. Choose a point x0 ∈ dom g and open sets U in X and W in Z, such that / Since g(U ∩Y ) ⊇ g(x0 ), we have that g(U ∩Y ) ∩ x0 ∈ U and W ∩ g(x0 ) 6= 0. W 6= 0/ and then g(U ∩Y ) ∩W 6= 0. / So, there is y0 ∈ U ∩Y , such that g(y0 ) ∈ W . Since Z is regular, there exists an open set W1 in Z, such that g(y0 ) ∈ W1 and W 1 ⊆ W . By the quasi-continuity of g, we conclude that there exists an open set U1 ⊆ U, such that U1 ∩Y 6= 0/ and g(U1 ∩Y ) ⊆ W1 . Fix x ∈ U1 . Then \

g(x) =

g(U ∩Y ) ⊆ g(U1 ∩Y ) ⊆ W 1 ⊆ W.

U∈Ux

Therefore, g is minimal.

t u

Proposition 11.3. Let X and Z be topological spaces, F : X ( Z be a minimal multifunction and f : X → Z be a function, such that f (x) ∈ F(x) for any x ∈ X. Then f is quasi-continuous. Proof. Fix x0 ∈ X. Let U and W by open sets, such that x0 ∈ X and f (x0 ) ∈ W . Since F is minimal and F(x0 ) ∩ W 3 f (x0 ), we have that there exists a nonempty open set U1 ∈ U with F(U1 ) ⊆ W . Therefore, f (U1 ) ⊆ F(U1 ) ⊆ W . t u Theorem 11.2. Let X be a topological space, Z be a compact, Y be a dense subset of X and g : Y → Z be a quasi-continuous function. Then there exists a e quasi-continuous function f : X → Z, such that f |Y = g and D( f ) = D(g). Proof. Since Z is a compact and Y = X, we have that dom g = X. Consider a function f : X → Z, such that f (x) ∈ g(x) for any x ∈ X and f (y) = g(y) for any y ∈ Y . By Propositions 11.2 and 11.3 we obtain, that f is a quasi-continuous extension of g.

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e Let us prove, that D( f ) = D(g). If x0 6∈ D( f ), then f is continuous at x0 . e So, lim g(y) = lim f (y) = lim f (x) = f (x0 ). Therefore, x0 6∈ D(g). In Y 3y→x0

Y 3y→x0

x→x0

e the other hand, if x0 6∈ D(g), then there exists z0 = lim g(y) ∈ Z. To prove Y 3y→x0

that x0 6∈ D( f ) it is sufficient to show that f (x) → z0 as x → x0 . Fix a neighborhood W of z0 . Then by regularity of the compact Z, there exists an open neighborhood W0 of z0 with W 0 ⊆ W . Since lim g(y) = z0 ∈ W0 , there is an Y 3y→x0

open neighborhood U0 of x0 , such that g(U0 ∩Y ) ⊆ W0 . So, for any x ∈ U0 we have that f (x) ∈ g(x) =

\

g(U ∩Y ) ⊆ g(U0 ∩Y ) ⊆ W 0 ⊆ W.

U∈Ux

Therefore, lim f (x) = z0 and then x0 6∈ D( f ). x→x0

t u

Corollary 11.1. Let X be a topological space, Z be a compact, Y be an open subset of X and g : Y → Z be a quasi-continuous function. Then there exists a quasi-continuous function f : X → Z, such that f |Y = g. Proof. By Theorem 11.2 there is a quasi-continuous extension h : Y → Z. Fix an arbitrary point z0 ∈ Z and define f : X → Z by the formula  h(x) , if x ∈ Y ; f (x) = z0 , if x ∈ X \Y . Obviously, f is an extension of g. Prove that f is quasi-continuous. Fix x ∈ X. In the case, where x ∈ X \ Y , the quasi-continuity f at x is evident. Consider the case, where x ∈ Y . Let U be an open neighborhood of x and W be a neighborhood of f (x) = h(x). The quasi-continuity of h implies that there is a nonempty set V1 such that V1 is open in Y , V1 ⊆ U and h(V1 ) ⊆ W . Put U1 = V1 ∩Y . Since Y is open and dense in Y , the set U1 is non-empty and open in X. But f (U1 ) = h(U1 ) ⊆ h(V1 ) ⊆ W . So, f is quasi-continuous at x. t u Proposition 11.4. Let Z be a non-compact topological space. Then there exist a topological space X, a dense subspace Y of X and a quasi-continuous function g : Y → Z which has no quasi-continuous extension f : X → Z. In the case, where Z is not countably compact we may assume that X = [0; 1] and Y = (0; 1]. Proof. Firstly, let us prove the second part of this proposition. Since Z is not countably compact, there exists a sequence (zn )∞ n=1 in Z which has no limits point in Z. Denote X = [0; 1] and Y = (0; 1]. Define a quasi-continuous function

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167

 1 1 g : Y → Z by the formula g(y) = zn for y ∈ n+1 ; n and n ∈ N. Let us prove, that g has no quasi-continuous extension. Assuming the contrary, let f : X → Z be a quasi-continuous extension of g. Let us prove that the point z0 = f (0) is the limit point of the sequence (zn )∞ n=1 . Let W be a neighborhood of z0 and n ∈ N. Since f is quasi-continuous at 0 and U = 0; n1 is a neighborhood of 0, there exist a non-empty open set U1 ⊆ U, such that f (U1 ) ⊆ W . Choose y ∈ U1 ∩ Y . Then f (y) = g(y) = zm for some m ≥ n. So, zm ∈ W . And then (zn ) has a limit point in Z, which is impossible. Now, let us prove the main part of the proposition. Assuming non-compactness of Z, we have that there exist a direct set (M, ≤) and a net (zm )m∈M without a limit point in Z. Let X = M ∪ {∞}, where ∞ is some element with ∞ 6∈ M. Put m ≤ ∞ for each m ∈ X. We equip X by the topology generating by the base   {m} : m ∈ M ∪ [m; ∞] : m ∈ M , where [m; ∞] = {n ∈ M : m ≤ n ≤ ∞}. Set Y = M. Define g : Y → Z by the formula g(m) = zm for any m ∈ M = Y . Since Y is discrete, g is continuous. Let us prove, that g is needed. Assuming the contrary, let f : X → Z be a quasi-continuous extension of g. Let us prove that the point z0 = f (∞) is the limit point of the net (zm )m∈M . Let W be a neighborhood of z0 and m0 ∈ M. Since f is quasi-continuous at ∞ and U = [m0 ; ∞] is a neighborhood of ∞, there exist a non-empty open set U1 ⊆ U, such that f (U1 ) ⊆ W . Choose m1 ∈ U1 ∩Y . Then m1 ≥ m0 and f (m1 ) = g(m1 ) = zm1 . So, zm1 ∈ W . And then (zm )m∈M has a limit point in Z, which is impossible. t u

11.4 Extension of real-valued quasi-continuous functions defined on a closed Baire subspace of hereditarily normal space Proposition 11.5. Let X be a hereditarily normal topological space, Y be a closed Baire subspace of X and g : Y → R be a quasi-continuous function. Then there exist a quasi-continuous function f : X → R such that f |Y = g, D( f ) = D(g) and sup | f (x)| = sup |g(y)|. x∈X

y∈Y

Proof. As it is well known (see for example [4, 7]), since f is quasi-continuous and Y is Baire space, we have that Y1 = C( f ) is dense in Y . Let g1 = g|Y1 . Then g1 is continuous and it is easy to show that sup |g1 (y)| = sup |g(y)|. By herediy∈Y1

y∈Y

tary normality of X, we have that subspace X1 = (X \Y )∪Y1 is normal. But Y1 is

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closed in X1 . So, by the Tietze-Urysohn theorem [1, p. 69] we construct a continuous function f1 : X1 → R such that f1 |Y1 = g1 and sup | f1 (x)| = sup |g1 (y)|. x∈X1

y∈Y1

Define f : X → R as  f (x) =

f1 (x), x ∈ X \Y ; g(x), x ∈ Y.

and prove that f is needed. Since f |X1 = f1 , we have that sup | f (x)| = sup |g(y)|. x∈X

y∈Y

Fix x0 ∈ X and prove that f is quasi-continuous at x0 . If x0 ∈ X \Y then this is imply by the continuity of f1 . So, suppose that x0 ∈ Y . Then f (x0 ) = g(x0 ). Let ε > 0 and U be an open neighborhood of x. Since g is quasi-continuous, there exists an open set U1 ⊆ U such that U ∩Y 6= 0/ and |g(y) − g(x0 )| < ε2 for any y ∈ U1 ∩Y . But Y1 is dense in Y . Thus, there is x1 ∈ U1 ∩Y1 . By continuity of f1 we find an open neighborhood U2 ⊆ U1 of x1 such that | f1 (x) − f1 (x1 )| < ε2 for all x ∈ U2 ∩X1 . But f1 (x1 ) = g(x1 ) and f1 (x) = f (x) for all x ∈ X \Y . Therefore, | f (x) − g(x1 )| < ε2 for each x ∈ U2 \Y . Then | f (x) − f (x0 )| = | f (x) − g(x0 )| ≤ | f (x) − g(x1 )| + |g(x1 ) − g(x0 )| < ε2 for all x ∈ U \ Y . But | f (x) − f (x0 )| = |g(x) − g(x0 )| < ε2 < ε for every x ∈ Y ∩ U2 . So, | f (x) − f (x0 )| < ε for all x ∈ U2 . Thus, f is quasi-continuous at x0 . Finally, let us prove that D( f ) = D(g). Obviously, D(g) = Y \C(g) = Y \Y1 and D(g) ⊆ D( f ). Since f is continuous on X \Y , we have that D( f ) ⊆ Y . For any y ∈ Y if y ∈ Y1 then g and f1 is continuous at y and, so, f is continuous at y. Thus, D( f ) ⊆ Y \Y1 = D(g). Therefore, D( f ) = D(g). t u Recall that a topological space X is called hereditarily Baire space if each closed subspace Y of X is a Baire space. From Propositions 11.1 and 11.5 we obtain the following result. Theorem 11.3. Let X be a hereditarily normal hereditarily Baire topological space, Y be a subspace of X and g : Y → R be a bounded quasi-continuous function. Then there exist a bounded quasi-continuous function f : X → R such e that f |Y = g, D( f ) = D(g) and sup | f (x)| = sup |g(y)|. x∈X

y∈Y

11.5 Universal extension of quasi-continuous functions Definition 11.6. Let X and Z be topological spaces, H be an open subspace of X, Y = X \ H. A function h : H → Z is called an universal quasi-continuous extension if for any function g : Y → Z, such that the restriction g|intY is quasicontinuous, the function f = g ∪ h : X → Z, which is defined by the formula

11. On extensions of quasi-continuous functions

 f (x) =

g(x) , if x ∈ Y ; h(x) , if x ∈ H,

169

for any x ∈ X,

is quasi-continuous. Firstly, we prove some simple characterization of universal quasi-continuous extensions. Proposition 11.6. Let X and Z be topological spaces, H be an open subspace of X, Y = X \ H and h : H → Z. Then the followings items are equivalent: (i) h is an universal quasi-continuous extension; (ii) h is quasi-continuous and the cluster set h(x) = Z for any x ∈ fr H; Proof. (i) ⇒ (ii). Prove, that h is quasi-continuous. Let g0 : Y → Z be a constant function. Then (i) implies that f0 = g0 ∪ h is quasi-continuous. Since H is open, the restriction h = f0 |H is quasi-continuous too. Fix z0 ∈ Z, x0 ∈ fr H and prove that z0 ∈ h(x0 ). Firstly, consider the case, where the only neighborhood of z0 is Z. Then z0 ∈ E for any non-empty set E ⊆ Z. So, z0 ∈ h(U ∩ H) for any neighborhood U of x0 . Therefore z0 ∈ h(x0 ). Now assume that there exists a neighborhood W0 of z0 , such W0 6= Z. Fix z1 ∈ Z \ W0 . Consider the function g : Y → Z such that g(y) = z0 for all y ∈ fr H = frY = Y \intY and g(y) = z1 for all y ∈ intY . By (i) we conclude that the function f = g ∪ h : X → Z is quasi-continuous. Consider open neighborhoods U of x0 and W of z0 . Since f (x0 ) = z0 and f is quasi-continuous, we have that there is a non-empty open set U1 ⊆ U, such that f (U1 ) ⊆ W ∩ W0 . But f (x) = z1 6∈ W0 for any x ∈ intY . So, U1 ⊆ X \ intY = H. Therefore, U1 ∩ H 6= 0. / Choose x1 ∈ U1 ∩ H. Then h(x1 ) = f (x1 ) ∈ W . Thus, W ∩ h(U ∩ H) 6= 0. / Therefore, z0 ∈ h(U ∩ H) for any U ∈ Ux0 . Then z0 ∈ h(x0 ). (ii) ⇒ (i). Let g : Y → Z be a function, such that the restriction g|intY is quasi-continuous, and let f = g ∪ h : X → Z. We have that the restrictions f |intY = g|intY and f |H = h are quasi-continuous. So, it is reminds to show, that f is quasi-continuous at every point of fr H. Fix x0 ∈ fr H. Consider open neighborhoods U of x0 and W of f (x0 ). Since f (x0 ) ∈ Z = h(x0 ) ⊆ h(U ∩ H), we have that W ∩ h(U ∩ H) 6= 0. / Choose x1 ∈ U ∩ H, such that h(x1 ) ∈ W . But h is quasi-continuous. So, there exists an non-empty open set U1 ⊆ U ∩ H with h(U1 ) ⊆ W . Then f (U1 ) = h(U1 ) ⊆ W . Therefore, f is quasi-continuous at x0 . t u The following theorem give some answer to the question on the existing of real-valued universal quasi-continuous extensions.

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Theorem 11.4. Let X be a perfectly normal space and H be an open subspace of X such that X0 = H is locally connected. Then there is a continuous function h : X \Y → R which is an universal quasi-continuous extension. Proof. Set U = X \ Y , V = intY , F = Y \ V . Then X0 = U ∪ F = X \ V and F is a nowhere dense subset of X0 . Let ϕ : (0; +∞) → R be a function which is defined by the formula ϕ(t) = 1t sin 1t for any t > 0. Obviously, ϕ is continuous and  ϕ (0; δ ) = R for all δ > 0. (11.1) By the Vedenissoff theorem [1, p. 45], there exists a continuous function ψ : X → [0; 1] such that ψ −1 (0) = F. Put h = ϕ ◦ ψ. Obviously, h is continuous. Prove that h is an universal quasi-continuous extension. Fix x0 ∈ F and prove that h(x0 ) = R. Let U be a neighborhood of x0 . By the local connectedness of X0 , there exists a connected neighborhood U1 of x0 in X0 , such that U1 ⊆ U. But F is nowhere dense in X0 . So, there exists a point x1 ∈ U1 \ F. Then ψ(x1 ) > 0 and ψ(x0 ) = 0. Let δ = ψ(x1 ). Since U1 is connected and ψ is continuous, we have that ψ(U1 ) ⊇ [0; δ ]. But ψ(x) = 0 on F and U1 ∩ H = U1 \ F. Therefore, ψ(U 1 ∩ H) ⊇ (0;δ ). Then  by(11.1) we conclude that h(U ∩ H) ⊇ h(U1 ∩ H) = ϕ ψ(U1 ∩ H) ⊇ ϕ (0; δ ) = R. Thus, h(x0 ) = R. Then Proposition 11.6 implies that h is an universal quasicontinuous extension. t u Proposition 11.5 and Theorem 11.4 imply the following. Corollary 11.2. Let X be a perfectly normal space, Y be a subspace of X, such that X0 = X \ intY is locally connected, and g : Y → R be a bounded quasicontinuous function. Then there is a quasi-continuous extension f : X → R of g.

11.6 Multicellular spaces and existing of universal quasicontinuous extensions Definition 11.7. For a topological space X the cardinal numbers d(X) = min{card A : A is a dense subset of X} and c(X) = sup{card U : U is a disjoint system of non-empty open subsets of X} are called the density and the cellularity of X respectively. We call a topological

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space X multicellular if there exists a disjoint family U of non-empty open subsets of X with cardinality card U = d(X). Clearly, c(X) ≤ d(X). Than c(X) = d(X) for any multicellular space X. But the inverse statement is not true. Proposition 11.7. If m is weakly inaccessible cardinal (i. e. regular and limit) and X = ∏n ℵ0 is regular. Fix some metric ρ generating the topology of X. Applying the Teichmüller-Tukey lemma [1, p. 8], we can choose for any n ∈ N a maximal set Tn , such that ρ(x, y) ≥ n1 for any distinct points x, y ∈ Sn . So, we have, that ρ(x, Tn )
ℵ0 . Therefore, n∈N

there exists n with mn = m. Then Un is a disjoint open family of cardinality m. Thus, X is multicellular. t u Lemma 11.1. Let X and Z be topological spaces, H be an open subspace of X, Y = X \ H, h : H → Z be an universal quasi-continuous extension and (Wt )t∈T be a disjoint family of non-empty open sets Wt in Z. Then there exists a disjoint family (Ht )t∈T of non-empty open sets Ht in H, such that fr H ⊆ H t for any t ∈ T. Proof. Denote Ht = int h−1 (Wt ) for t ∈ T . Obviously, the sets Ht are disjoint and open. Fix t ∈ T and x ∈ fr H. To prove that x ∈ H t consider an open neighborhood U of x. By Proposition 11.6 we have that h is quasicontinuous and the cluster set h(x) = Z. Thus Wt ∩ h(H ∩U) ⊇ Wt ∩ h(x) 6= 0. / Then Wt ∩ h(H ∩ U) 6= 0. / Therefore, the quasi-continuity of h implies that there is a non-empty open subset U1 of H ∩ U such that h(U1 ) ⊆ Wt . So, U ∩ Ht = U ∩ int h−1 (Wt ) ⊇ U1 6= 0. / t u Lemma 11.2. Let X be a topological spaces, Z be a compact, H be an open subspace of X, (Ht )t∈T be a disjoint family of non-empty open sets Ht in H, such that fr H ⊆ H t for any t ∈ T and {zt : t ∈ T } be a dense subset of Z. Then there exists an universal quasi-continuous extension h : H → Z. Proof. Let G =

S

Ht . Define g : G → Z by the formula g(x) = zt if x ∈ Ht for

t∈T

some t ∈ T . Since fr H ⊆ H t for any t ∈ T , we conclude that fr H ⊆ G. Fix x ∈ fr H and prove that g(x) = Z. Let z ∈ Z, U is a neighborhood of x and W is a neighborhood of z. Since {zt : t ∈ T } = Z, we can find t ∈ T such that zt ∈ W . But U ∩ Ht 6= 0. / Thus, g(U ∩ H) ⊇ g(U ∩ Ht ) = {zt }. Therefore,

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g(U ∩ H) ∩ W 6= 0/ for any neighborhood W of z. So, z ∈ g(U ∩ H) for any S g(U ∩ H) = g(x) for any z ∈ Z. Therefore, neighborhood U of x. Then z ∈ U∈Ux

g(x) = Z. By Corollary 11.1 we can construct a quasi-continuous extension h : H → Z of the function g. Then for any x ∈ fr H we have that Z ⊆ g(x) ⊆ h(x). So, h(x) = Z for such x. Thus, Proposition 11.6 implies that h is an universal quasicontinuous extension. t u From the Lemmas 11.1 and 11.2 we immediately obtain the following. Theorem 11.5. Let X be a topological space, Z be a multicellular compact, m = d(Z), H be an open subset of X. Then the following conditions are equivalent (i) there exists an universal quasi-continuous extension h : H → Z; (ii) there exists a disjoint family (Ht )t∈T of open subsets Ht of H, such that card T = m and fr H ⊆ H t for any t ∈ T . Using Proposition 11.9 we conclude from previous theorem the following. Theorem 11.6. Let X be a topological space, Z be an infinity separable compact, H be an open subset of X. Then the following conditions are equivalent (i) there exists an universal quasi-continuous extension h : H → Z; (ii) there exists a disjoint family (Hn )∞ n=1 of non-empty open sets Hn in H, such that fr H ⊆ H n for any n ∈ N. Definition 11.8. A topological space X is called weakly pairwise attainable if for any open set G and any closed subset F of G \ G there exist disjoint opens subsets U and V of G such that F = U \ G = V \ G. In [6] it was proved the following: if X is a perfectly normal FréchetUrysohn space such that for any closed subset F of X there is a strongly σ discrete set E with E = F then X is weakly pairwise attainable. In particular, in [6] (see also [5]) it was proved that every metrizable space is weakly pairwise attainable. Theorem 11.7. Let X be a weakly pairwise attainable space, Z be a separable compact, H be an open subset of X. Then there exists an universal quasicontinuous extension h : H → Z. Proof. Let F = fr H. Since X is weakly pairwise attainable, there are disjoint opens subsets H1 and G1 of H such that F = H 1 \ H = G1 \ H. In particular,

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F ⊆ fr G1 . Then there are disjoint opens subsets H2 and G2 of G1 such that F = H 2 \ G1 = G2 \ G2 . So, we can construct a sequence of nonempty open sets Gn and Hn such that G1 t H1 ⊆ H, Gn+1 t Hn+1 ⊆ Gn and F ⊆ Gn ∩ H n for any n ∈ N. Therefore, the sets Hn are pairwise disjoint and F ⊆ H n for each n. By Proposition 11.9 Z is multicellular and ν = d(X) ≤ ℵ0 . Put T = N if ν = ℵ0 and T = {1, 2, . . . ν} if ν < ℵ0 . Then (Hn )n∈T satisfies the condition (ii) from Theorem 11.5. Thus, there exists an universal quasi-continuous extension h : H → Z. t u The Theorems 11.7 and 11.2 immediately imply the following. Corollary 11.3. Let X be a weakly pairwise attainable space, Z be a separable compact, Y ⊆ X and g : Y → Z be a quasi-continuous function. Then there exist a quasi-continuous function f : X → Z, such that f |Y = g.

11.7 Quasi-open, quasi-closed and quasi-clopen sets Definition 11.9. Let X be a topological space and A ⊆ X. A set A is called • • • • •

quasi-open if A ⊆ intA; quasi-closed if intA ⊆ A; quasi-clopen if intA ⊆ A ⊆ intA; regularly open if A = intA; regularly closed if A = int A.

It is easy to see that a function f : X → Y between topological spaces X and Y is quasi-continuous if and only if f −1 (V ) is quasi-open in X for each open set V in Y . In [3] quasi-open sets and quasi-continuous functions are called semiopen and semi-continuous respectively. But in modern articles such functions f is always called quasi-continuous. So, we use term “quasi-open” for such sets. Regularly open (closed) sets are sometimes called open (closed) domain (see [1, p. 20]) or canonically open (closed). Quasi-clopen sets sometimes are called canonical. Proposition 11.11. Let X be a topological space and A ⊆ X. Then (i) (ii) (iii) (iv) (v)

A is quasi-open if and only if X \ A is quasi-closed; A is quasi-closed if and only if X \ A is quasi-open; if A is open then A is quasi-open; if A is closed then A is quasi-closed; A is quasi-clopen if and only if A is both quasi-open and quasi-closed;

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175

(vi) A is regularly open if and only if A is both quasi-clopen and open; (vii) A is regularly closed if and only if A is both quasi-clopen and closed; (viii)A is quasi-clopen if and only if X \ A is quasi-clopen; (ix) if A is closed then intA is regularly open; (x) if A is open then A is regularly closed. Proof. (i) As it is well known, for any E ⊆ X we have that int(X \ E) = X \ E

and

X \ E = X \ intE.

By definition of quasi-open sets we have that A ⊆ intA. Put B = X \ A. So, B = X \ A ⊇ X \ intA = int(X \ intA) = intX \ A = intB. Therefore B is quasi-closed. (ii) Obviously, (ii) ⇔ (i). (iii) Let A be open. Then A = intA ⊆ intA. So, A is quasi-open. (iv) Let A be closed. Then A = A ⊇ intA. So, A is quasi-closed. (v) This item is evident. (vi) Let A be a regularly open subset of X. Then A = intA. So, A is quasiclosed. Openness of the interior implies that A is open. Thus, by (iii), A is quasi-open. Then, by (v), A is quasi-clopen. Let A be an open subset of X which is quasi-clopen. Then A is quasi-closed. Thus, intA ⊆ A = intA ⊆ intA. Therefore, intA = A. So, A is regularly open. (vii) Let A be a regularly closed subset of X. Then A = intA. So, A is quasiopen. Closeness of the closure implies that A is closed. Thus, by (iv), A is quasi-closed. Then, by (v), A is quasi-clopen. Let A be a closed subset of X which is quasi-clopen. Then A is quasi-open. Thus, intA ⊇ A = A ⊇ intA. Therefore, intA = A. So, A is regularly closed. (viii) This item follows from (i), (ii) and (v). (ix) Let B = intA. Then intB ⊆ B ⊆ A = A. So, intB ⊆ intA = B. But B = intB ⊆ intB. Therefore, B = intB, that is B is regularly open. (x) Put B = A. Then intB ⊇ intB ⊇ intA = A. So, intB ⊇ A = B. But B = B ⊇ intB. Therefore, B = intB, that is B is regularly closed. t u Proposition 11.12. Let X be a topological space and A ⊆ X. Then A is quasi-open if and only if there is an open set U and a set E ⊆ U \U such that A = U ∪ E; (ii) A is quasi-closed if and only if there is a closed set F and a set E ⊆ F \ int F such that A = F \ E;

(i)

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(iii) A is quasi-open if and only if there exists an regularly closed set F and a nowhere dense set E ⊆ F such that A = F \ E; (iv) A is quasi-closed if and only if there exists an regularly open set U and a nowhere dense set E ⊆ X \U such that A = U ∪ E; (v) A is quasi-clopen if and only if there is a regularly open set U and a set E ⊆ U \U such that A = U ∪ E; (vi) A is quasi-clopen if and only if there is a regularly closed set F and a set E ⊆ F \ int F such that A = F \ E. Proof. (i) Let A be a quasi-open subset of X. Put U = intA and E = A \ U. Then A = U ∪ E. By the definition of quasi-open sets, we have that A ⊆ U. So, E = A \U ⊆ U \U. Therefore, U and E are to be found. Conversely, let A = U ∪ E for some open set U and set E ⊆ U \ U. Since U = intU ⊆ intA, we conclude that A = U ∪ E ⊆ U ⊆ intA. Thus, A is quasiopen. (ii) Let A be a quasi-closed subset of X. Put F = A and E = F \ A. Then A = F \ E. By the definition of quasi-closed sets, we have that intF ⊆ A. So, E = F \ A ⊆ F \ intF. Therefore, F and E are to be found. Conversely, let A = F \ E for some closed set F and set E ⊆ F \ intF. Since F = F ⊇ A, we conclude that A = F \ E ⊇ F \ (F \ intF) = intF ⊇ intA. Thus, A is quasi-closed. (iii) Let U = int A. Since A is quasi-open, A ⊆ U. Proposition 11.11(x) yields that F = U is regularly closed. Set E = F \ A. Then A = F \ E, E ⊆ U \U = frU is nowhere dense. So, F and E is to be found. (iv) Let F = A. Since A is quasi-closed, int F ⊆ A. Proposition 11.11(ix) yields that U = int F is regularly closed. Set E = A \ U. Then A = U ∪ E, E ⊆ F \ int F = fr F is nowhere dense. So, U and E is to be found. (v) and (vi) Let A be a quasi-clopen subset of X. As in (i) and (ii), put U = intA, F = A and E 0 = A \ U and E 00 = F \ A. Then A = U ∪ E 0 = F \ E 00 , E 0 ⊆ U \U and E 00 ⊆ F \ intF. Let us prove that U is regularly open and F is regularly closed. Taking to account that U ⊆ F and by the definition of quasiclopen subsets intF ⊆ A ⊆ U, we obtain that intF = int(intF) ⊆ intA = U = intU ⊆ intF and U ⊆ F = F = A ⊆ (U) = U. Thus, intF = U and U = F. Therefore, intF = U = F and intU = intF = U. So, U is regularly open and F is regularly closed. Let A = U ∪ E where U is regularly open subset of X and E ⊆ U \U. Then by (i) we have that A is quasi-open. To prove that A is quasi-closed, observe that intA = int(U ∪ E) = intU = U ⊆ A. Then by Proposition 11.13(v) we have that A is quasi-clopen.

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177

Let A = F \E where F is regularly closed subset of X and E ⊆ F \intF. Then by (ii) we have that A is quasi-closed. Since F \ intF is closed, we have that E ⊆ F \ intF and then E ∩ intF = 0. / So, intA = int(F \ E) = int(F ∩ (X \ E)) = intF ∩ int(X \ E) = intF ∩ (X \ E) = intF \ E = intF. Thus, intA = intF = F ⊇ A. Therefore, A is quasi-open. Then by Proposition 11.13(v) we have that A is quasi-clopen. t u Proposition 11.13. Let X be a topological space and B ⊆ A ⊆ X. Then (i)

if A is quasi-open in X and B is quasi-open in A then B is quasi-open in X; (ii) if A is quasi-closed in X and B is quasi-closed in A then B is quasi-closed in X; (iii) if A is quasi-clopen in X and B is quasi-clopen in A then B is quasi-clopen in X. Proof. (i) Put U = intA and V = intA B. Then A ⊆ U and B ⊆ V . Since V is open in A and U is dense in A, we have that the set W = U ∩V is dense in V . So, W = V . On the other hand, W is open in U and U is open in X. Therefore, W is open in X. So, W ⊆ intB. Then B ⊆ W ⊆ intB. (ii) Put U = intA and V = intA (B ∩ A). Then U ⊆ A and V ⊆ B. Since V is open in U, the set W = U ∩W is open in U. But U is open in X. Therefore W is open in X. Let us prove that E = A\U is nowhere dense in X. Indeed, E ⊆ X \U implies intE ⊆ E ⊆ X \U = X \ U. In the other hand, intE ⊆ intA = U. Therefore intE = 0 and then E is nowhere dense. Analogously, the set E1 = B \ V is nowhere dense in A. Since U is open, E1 ∩ U is nowhere dense in U. Therefore E1 ∩ U is nowhere dense in X. But E1 \ U is nowhere dense in X as a subset of the nowhere dense set A \ U. So, E1 is nowhere dense in X, because E1 = (E1 \U) ∪ (E1 ∩U). Set G = intW and prove that G = intB. Clearly, W ⊆ V ⊆ B. Then G = intW ⊆ intB. To prove the inverse inclusion, we observe that B \W is nowhere dense in X, because B \ W ⊆ B \ W = (B \ U) ∪ (B \ V ) ⊆ E ∪ E1 . But X \ W is open set. So, B \ W = B ∩ (X \ W ) ⊆ B ∩ (X \W ) = B \W . Thus, B \ W is nowhere dense in X. So, intB ⊆ W . Therefore, intB ⊆ intW = G. Thus, we have proved that intB = G. (iii) This item follows from (i), (ii) and Proposition 11.11(v). t u

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The following proposition implies straitly from the definitions. Proposition 11.14. Let X and Y be topological spaces and f : X → Y . Then the following statement are equivalent: (i) f is quasi-continuous; (ii) for any open subset G of Y the pre-image f −1 (G) is quasi-open; (iii) for any closed subset F of Y the pre-image f −1 (F) is quasi-closed.

11.8 Quasi-clopen partitions and quasi-continuous functions We need some auxiliary statements. Lemma 11.3. Let X be a hereditarily normal space, Y ⊆ X and B be a quasiclopen subset of Y . Then there exist a quasi-clopen subset A of X with A ∩Y = B. Y

Y

Proof. Put V = intY B and V 0 = Y \ B . Since B is quasi-clopen in Y , we have Y Y Y Y that V ⊆ B ⊆ V . In particular, B = V . Thus, V = intY V . Obviously, V ∩ Y Y V 0 = V ∩V 0 = 0. / So, V ∩V 0 = V ∩V 0 = 0, / that is V and V 0 are separated in X. By [1, Theorem 2.1.7.] we have that there exist open subsets U and U 0 of X, such that V ⊆ U, V 0 ⊆ U 0 and U ∩U 0 = 0. / Then V ⊆ U ∩Y ⊆ (X \ U 0 ) ∩Y = Y Y Y Y \ (U 0 ∩ Y ) ⊆ Y \ V 0 = B = V . So, V ⊆ U ∩ Y ⊆ intY V = V . Therefore, U ∩Y = V . Put F = U. By Proposition 11.11(ix), F is a regularly open subset Y of X. Observe, that intY V = V = U ∩Y ⊆ intF ∩Y ⊆ F ∩Y ⊆ (X \U 0 ) ∩Y ⊆ Y Y Y Y \V 0 = V . So, int F ∩Y = V and F ∩Y = V . Let E = V \ B and A = F \ E. Y Then E ⊆ V \V = (F ∩Y ) \ (int F ∩Y ) ⊆ F \ int F. By Proposition 11.13(vi), we have that B is quasi-clopen. Observe, that A ∩Y = (F \ E) ∩Y = (F ∩Y ) \ Y Y E = V \ (V \ B) = B. Therefore, A is to be found. t u Definition 11.10. A system A of subsets of a topological space X is called a quasi-clopen partition of X if A is a finite system of pairwise disjoint nonS empty quasi-clopen subsets of X, such that A = X. Lemma 11.4. Let X be a hereditarily normal space, Y ⊆ X and B be a quasiclopen partition of Y . Then there exist a quasi-clopen partition A of X such that B = {A ∩Y : A ∈ A }. Proof. Prove this lemma by induction on card B. If cardB = 0 then B = 0/ and A = 0/ is needed. Suppose, that for some n > 0 we have, that the lemma

11. On extensions of quasi-continuous functions

179

holds for any B with card B = n − 1. Let card B = n. Since n > 0, there exists B1 ∈ B. By Lemma 11.3, there exists a quasi-clopen in X set A1 , such that A1 ∩ Y = B1 . Put X 0 = X \ A1 , Y 0 = Y \ B1 and B 0 = B \ {B1 }. Then B 0 is a quasi-clopen partition of a subspace Y 0 of a hereditarily normal space X 0 with cardB 0 = n − 1. By the inductive assumption, we have that there exists a quasi-clopen partition A 0 of X 0 , such that B 0 = {A ∩Y 0 : A ∈ A 0 }. By Proposition 11.13(viii), Y 0 is quasi-clopen in X. Then Proposition 11.13(iii) implies, that for any A ∈ A 0 , we have that A is quasi-clopen in X. Therefore, A = A 0 ∪ {A1 } is needed. t u Lemma 11.5. Let Y be a topological space, Z be a metric compact, g : Y → Z be a quasi-continuous function and ε > 0. Then there exists a quasi-clopen partition B of Y such that diam g(B) < ε for all B ∈ B. Proof. Let nε (Z) be the minimal of numbers n ∈ N for which there exists a finite open covering {W1 ,W2 , . . . ,Wn } of Z, such that diamWi < ε for all i = 1, 2, . . . , n. Prove our statement by the induction on nε (Z). If nε (Z) = 1, then diam Z < ε. So, B = {Y } \ {0} / is needed. Suppose, that for some n ∈ N the lemma holds for any Y , Z and g with nε (Z) < n. Fix some Y , Z, g with nε (Z) = n and prove the existing of B. Choose a finite open covering {W1 ,W2 , . . . ,Wn } of Z, such that diamWi < ε for all i = 1, 2, . . . , n. Set An = g−1 (Wn ), Mn = g−1 (W n ). By Proposition 11.14, we conclude that An is quasi-open and Mn is quasi-closed. Thus, An ⊆ int An and int M n ⊆ Mn . Let Un = int An . Put Bn = Mn ∩U n . Then g(Bn ) = g(Mn ) ⊆ W n = W n . So, diam g(Bn ) ≤ diamW n = diamWn < ε. Since An ⊆ Mn , we have that Un ⊆ int M n ⊆ Mn . Thus, Un ⊆ U n ∩ Mn = Bn ⊆ U n . By Proposition 11.11(ix), Un is regularly open. Let En = Mn \ Un . Then En ⊆ U n \Un and Bn = Un ∪ En . By Proposition 11.13(v), Bn is quasi-clopen. Put Y 0 = Y \ Bn , Z 0 = Z \Wn and g0 = g|Y 0 . Clearly, nε (Z 0 ) ≤ n −1. By Proposition 11.11(viii), Y 0 is quasi-clopen. Let us prove, that g0 is quasi-continuous. Fix y0 ∈ Y 0 . Consider open sets V in Y and W in Z, such that y0 ∈ V and f (y0 ) ∈ W . Since y0 6∈ Bn = U n ∩ Mn , we conclude that y0 6∈ U n or y0 6∈ Mn . Firstly, consider the case, where y0 6∈ U n . By quasi-continuity of g, we have that there exists nonempty open sets U ⊆ V \ U n with g(U) ⊆ W . Since Bn ⊆ U n , we obtain that U ⊆ Y \ U n ⊆ Y \ Bn = Y 0 . Then g0 is quasi-continuous at y0 . Now, consider the case, where y0 6∈ Mn . Then g(y0 ) 6∈ W n . By quasicontinuity of g, we have that there exists a nonempty open set U ⊆ V with g(U) ⊆ W \W n . Then g(U) ⊆ Z \W n . Thus, U ⊆ g−1 (Z \W n ) = Y \g−1 (W n ) = Y \ Mn ⊆ Y \ Bn = Y 0 . Then g0 is quasi-continuous at y0 .

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Clearly, Y 0 , Z 0 and g0 satisfy all the required conditions and nε (Z 0 ) < n. Then, by the inductive assumption, we conclude that there exist a quasi-clopen partition B 0 of Y 0 , such that diam g0 (B) < ε for any B ∈ B 0 . By Proposition 11.13(iii), since Y 0 is quasi-clopen in Y , we have that B is quasi-clopen in Y for every B ∈ B 0 . Then B = B 0 ∪ ({Bn } \ {0}) / is to be found. t u The following lemma follows immediately from Lemmas 11.4 and 11.5. Lemma 11.6. Let X be a hereditarily normal space, Y ⊆ X, Z be a metric compact, g : Y → Z be a quasi-continuous function and ε > 0. Then there exists a quasi-clopen partition A of X such that B = {A ∩ Y : A ∈ A } is a quasiclopen partition of Y and diam g(B) < ε for all B ∈ B. Lemma 11.7. Let X be a topological spaces, Y be a T1 -space and f : X → Y be a function with the finite range f (X). Then f is quasi-continuous if and only if A = { f −1 (y) : y ∈ f (X)} is a quasi-clopen partition of X.

11.9 Extension of quasi-continuous function ranged in a metrizable compact Theorem 11.8. Let X be a hereditarily normal space, Y ⊆ X, Z be a metrizable compact, g : Y → Z be a quasi-continuous function. Then there exists a quasicontinuous function f : X → Z such that f |Y = g. Proof. Consider some metric d on the topological space Z for which diam Z < 1. We will construct a sequence (An )∞ n=1 such that for all n ∈ N the following conditions are hold: A1 = {X}; (11.2) for all A ∈ An+1 there exists B ∈ An such that A ⊆ B; An is a quasi-clopen partition of X; Bn = {A ∩Y : A ∈ An } is a quasi-clopen partition of Y ; diam g(B)
0 and 0 ε})] > 0 (12.3)

{nm } ε>0 (a,b)⊂[0,1] x∈(a,b) m→∞

and ∀







lim [dist(x, Op{t : fnm (t) − f (t) < −ε})] > 0. (12.4)

{nm } ε>0 (a,b)⊂[0,1] x∈(a,b) m→∞

From the supposition that { fn }n∈N does not converge to f we conclude that there exists a subsequence { fnm }m∈N without a subsequence { fnm p } p∈N convergent I -a.e. to f . Take this subsequence { fnm }m∈N and fix ε > 0. Let {(ai , bi )}i∈N be a basis of (0, 1) in the natural topology. From (12.3) there exists a point x10 ∈ (a1 , b1 ) such that lim [dist(x10 , Op{t : fnm (t) − f (t) > ε})] > 0,

m→∞

(1)0

so there exist a subsequence {nm } of {nm } and an interval (x10 − δ10 , x10 + δ10 ) ⊂ (a1 , b1 ) such that for each m ∈ N, (x10 − δ10 , x10 + δ10 ) ∩ Op{t : f

(1)0

nm

(t) − f (t) > ε} = 0. /

(1)0

From (12.4) for {nm }, the same ε and for the interval (x10 − δ10 , x10 + δ10 ) sim(1) (1)0 ilarly we find a point x1 ∈ (x10 − δ10 , x10 + δ10 ), a subsequence {nm } of {nm } and an interval (x1 − δ1 , x1 + δ1 ) ⊂ (x10 − δ10 , x10 + δ10 ) ⊂ (a1 , b1 ) such that for each m ∈ N, (x1 − δ1 , x1 + δ1 ) ∩ Op{t : fn(1) (t) − f (t) < −ε} = 0. / m

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So finally we have (x1 − δ1 , x1 + δ1 ) ∩ Op{t : | fn(1) (t) − f (t)| > ε} = 0/ m

for each m ∈ N (because Op(A ∩ B) differs from Op(A) ∩ Op(B) by a first category set for A, B having the Baire property). (i) Suppose that for some i ∈ N we have chosen a subsequence {nm } of {nm } and the interval (xi − δi , xi + δ1 ) ⊂ (ai , bi ) such that (xi − δi , xi + δi ) ∩ Op{t : | fn(i) (t) − f (t)| > ε} = 0/ m

for each m ∈ N. (i) From (12.3) for {nm }, the same ε and for the interval (ai+1 , bi+1 ) there exist (i+1)0 0 − δ 0 , x0 + δ 0 ) ⊂ (a a subsequence {nm } and an interval (xi+1 i+1 , bi+1 ) i+1 i+1 i+1 such that 0 0 0 0 (xi+1 − δi+1 , xi+1 + δi+1 ) ∩ Op{t : f

(i+1)0

nm

(t) − f (t) > ε} = 0/

(i+1)0

for each m ∈ N. From (12.4) for {nm }, the same ε and for the interval (i+1) 0 0 , x0 0 (xi+1 − δi+1 } and an interval i+1 + δi+1 ) there exist a subsequence {nm 0 0 0 0 (xi+1 − δi+1 , xi+1 + δi+1 ) ⊂ (xi+1 − δi+1 , xi+1 + δi+1 ) such that (xi+1 − δi+1 , xi+1 + δi+1 ) ∩ Op{t : fn(i+1) (t) − f (t) < −ε} = 0/ m

for each m ∈ N. So finally we have (xi+1 − δi+1 , xi+1 + δi+1 ) ∩ Op{t : | fn(i+1) (t) − f (t)| > ε} = 0/ m

for each m ∈ N (and also (xi+1 − δi+1 , xi+1 + δi+1 ) ⊂ (ai+1 , bi+1 )). Now let us consider a decreasing sequence {{ fn(i) }m∈N }i∈N of subsequences m of { fnm }m∈N and let {gεm }m∈N be a diagonal sequence (i.e. gεm = fn(m) ). We shall m show that lim supm {t : |gεm (t)− f (t)| > ε} = Eε is of the first category. Suppose that this is not the case. Then there exists an interval (c, d) ⊂ [0, 1] such that Eε is residual on this interval (Eε has the Baire property). Let i0 be a natural number for which (ai0 , bi0 ) ⊂ (c, d). Then we have (xi0 − δi0 , xi0 + δi0 ) ∩ Op{t : |gεm (t) − f (t)| > ε} = 0/ for almost all m ∈ N (m ≥ i0 ). Hence (xi0 − δi0 , xi0 + δi0 ) ∩ Eε is a set of the first category – a contradiction, because (xi0 − δi0 + δi+0 ) ⊂ (ai0 , bi0 ) ⊂ (c, d).

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189

Now suppose that ε = 1. By virtue of the above reasoning we obtain a subsequence {g1m }m∈N of { fnm }. Repeat the argument for ε = 12 and {g1m } to obtain 1

a subsequence {gm2 }m∈N of {g1m } and proceed further by induction. From the 1

decreasing sequence {{gmk }m∈N }k∈N of subsequence { fnm } we choose a diago1

nal sequence {gm }m∈M (i.e. gm = gmm ). We shall show that {gm }m∈N converges S I -a.e. to f . Indeed, let E = k∈N E 1 , (E 1 is Eε for ε = 1k ). The set E is of the k

k

first category. Observe that Fk = lim supm {t : |gm (t) − f (t)| > 1k } ⊂ E 1 because k

1

{gm }m∈N is almost a subsequence of {gmk }. Hence E0 = the first category. Observe that [0, 1] \ E0 =

\

k∈N Fk ⊂ E is also of

S

([0, 1] \ Fk ) =

k∈N

=

\

  lim inf [0, 1] \ t : |gm (t) − g(t)| > 1k = m

k∈N

=

\

  lim inf t : |gm (t) − g(t)| ≤ 1k m

k∈N

is a residual set and for x ∈ [0, 1] \ E0 we have gm (x) −→ f (x). It is a conm→∞

tradiction because {gm }m∈N is a subsequence of { fnm } and {gm } should not converge I -a.e. to f . So we proved the existence of {nm }, ε0 > 0 and (a, b) ⊂ [0, 1] such that (12.1) or (12.2) holds. We shall consider the first case. In the second the argument is similar. Observe first that if for {nm }, (a, b) and ε0 > 0 we have the following property: for each x ∈ (a, b), lim [dist(x, Op{t : fnm (t) − f (t) > ε0 })] = 0, m→∞

then also for each x ∈ (a, b), lim [dist(x, (a, b) ∩ Op{t : fnm (t) − f (t) > ε0 })] = 0.

m→∞

(12.5)

Observe also that if G ⊂ (a, b) is an open set, G = ∞j=1 (a j , b j ) and for some δ > 0 we have dist(x, G) < δ for each x ∈ (a, b), then there exists j0 ∈ N such that dist(x, (a j0 , b j0 )) < δ for each x ∈ (a, b). Subtracting, if necessary, a finite set, we obtain an open set G0 ⊂ G with the following properties: S

190

a) b) c)

Władysław Wilczy´nski

G0 has a finite number of components, the length of each component of G0 is less than δ , dist(x, G0 ) < δ for each x ∈ (a, b).

Now we shall choose from { fnm } a subsequence { fnm p } p∈N which does not include a subsequence with convex combinations of the required form convergent I -a.e. to f (which will end the whole proof). Put δ1 = 31 . From (12.5) we conclude that there exists nm1 ∈ N such that for each x ∈ (a, b), dist(x, (a, b) ∩ Op{t : fnm1 (t) − f (t) > ε0 }) < δ1 (this is for all x simultaneously, which easily follows from (12.5) by using finite set {x1 , . . . , xk } ⊂ (a, b) forming a δ21 net). Let G1 ⊂ (a, b) ∩ Op{t : fnm1 (t) − f (t) > ε0 } be an open set with properties a), b), c) for δ1 , i.e. G1 has a finite number of components, each of length less than δ1 and such that dist(x, G1 ) < δ1 for each x ∈ (a, b). Let ∆1 be the length of the shortest component of G1 . Obviously ∆1 < δ1 . Put δ2 = ∆41 . Let nm2 > nm1 be such natural number that dist(x, (a, b) ∩ Op{t : fnm2 (t) − f (t) > ε0 }) < δ2 for each x ∈ (a, b) (the existence follows from (12.5) again). Choose an open set G2 ⊂ (a, b) ∩ Op{t : fnm2 (t) − f (t) > ε0 } having a finite number of components, each of length less than δ2 and such that dist(x, G2 ) < δ2 for each x ∈ (a, b). Observe that from the fact that δ2 = ∆41 if follows that each component of G1 includes some component of G2 . Suppose that we have defined an increasing finite subsequence nm1 , . . . , nm p of natural numbers and finite sequence of open sets G1 , . . . , G p such that each has only finite number of components, Gi ⊂ (a, b)∩Op{t : fnmi (t)− f (t) > ε0 }, dist(x, Gi ) < δi for each x ∈ (a, b) and the length of each component of Gi is less than δi for i = 1, . . . , p, moreover δi+1 < δ4i for i = 1, . . . , p − 1. Let ∆ p be the length of the shortest component of G p . Obviously ∆ p < δ p . ∆ Put δ p+1 = 4p . Let nm p+1 > nm p be such natural number that dist(x, (a, b) ∩ Op{t : fnm p+1 (t) − f (t) > ε0 }) < δ p+1 for each x ∈ (a, b).

12. Weak convergence with respect to category

191

Choose an open set G p+1 ⊂ (a, b) ∩ Op{t : fnm p+1 (t) − f (t) > ε0 } having a finite number of components, each of length less than δ p+1 and such that dist(x, G p+1 ) < δ p+1 for each x ∈ (a, b). Observe that from the fact that δ p+1 = ∆p 4 it follows that each component of G p includes some component of G p+1 . Thus by induction we have defined an increasing sequence {nm p } p∈N of natural numbers and the sequence {G p } p∈N of open sets fulfilling the following conditions: (i) (ii) (iii) (iv)

for each p ∈ N G p ⊂ (a, b) ∩ Op{t : fnm p (t) − f (t) > ε0 }; dist(x, G p ) < 41p for each p ∈ N and for each x ∈ (a, b); for each p ∈ N each component of G p has length less than 41p ; for each p ∈ N each component of G p includes some component of G p+1 .

p+k ai fnmi be a convex combination. We have Let φ = ∑i=p

Op{t : φ (t) − f (t) > ε0 } ⊃

p+k \

(Gi \ Pi ) = H,

i=p

where Pi are sets of the first category. From (iv) it follows that each component of G p includes some component of Op(H). From (ii) and (iii) it follows that dist(x, Op(H)) < 42p for each x ∈ (a, b). If we take an arbitrary subsequence { fnm pr } of { fnm p } and an arbitrary ser quence {φr } of convex combinations of the required form, i.e. φr = ∑ki=0 ar,i · fnm p and if Hr is a set described above attached to φr , then we have r+1

lim sup φr (t) − f (t) > ε0 r

I -a.e. on the set A = r=1 i=r Hi . But ∞ i=r Op(Hi ) is an open set dense on (a, b), so A is residual on (a, b). This means that { fn } is not weakly convergent to f with respect to I . t u T∞ S∞

S

References [1] A. Alexiewicz, Functional Analysis, MM, T. 49, Warszawa 1969 (in Polish). [2] N. Dunford, J. T. Schwartz, Linear Operators, Part I, Interscience Publishers 1958. [3] W. Poreda, E. Wagner-Bojakowska,W. Wilczy´nski, A category analogue of the density topology, Fund. Math. 125 (1985), 167-173. [4] E. Wagner, Sequences of measurable functions, Fund. Math. 112 (1981), 89-102.

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[5] W. Wilczy´nski, Problem Concerning the Lebesgue Density at a Point, Unsolved Problems on Mathematics for the 21st Century, A Tribute to Kiyoshi Iseki’s 80th Birthday, Jair Minoro Abe and Shotaro Tanaka (editors), IOS Press, 2001.

´ W ŁADYSŁAW W ILCZY NSKI Department of Mathematics and Computer Science, Łód´z University ul. S. Banacha 22, 90-238 Łód´z, Poland E-mail: wwil@uni.lodz.pl

Chapter 13

New properties of the families of convergent and divergent permutations - Part I

ROMAN WITUŁA, EDYTA HETMANIOK AND DAMIAN SŁOTA

2010 Mathematics Subject Classification: 40A05, 05A99. Key words and phrases: convergent permutations, divergent permutations.

13.1 Introduction In this paper, the permutations p of N are divided into two types, namely, the permutations preserving convergence of rearranged real series (i.e. the ones for which the series ∑ a p(n) is convergent for any convergent real series ∑ an ) and the other permutations, without this property. The first ones will be called the convergent permutations, the second ones – the divergent permutations (so for any divergent permutation p of N there exists a convergent real series ∑ an such that the p−rearranged series ∑ a p(n) is divergent). Convergent permutations have been characterized by many authors and in many ways. The following, very illustrative description is given by Agnew [1]: Theorem 13.1. A permutation p of N is the convergent permutation if and only if there exists a positive integer n such that for any interval I of N (i.e. a subset of N having the form {k, k + 1, . . . , k + m − 1} for some k, m ∈ N) the set p(I) is a union of at most n mutually separated intervals (abbrev.: n MSI). We have noticed that, on the ground of the above theorem, one can formulate the following dual characterization of the divergent permutations:

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Theorem 13.2. A permutation p of N is the divergent permutation if and only if for every n ∈ N there exists an interval I of N such that the set p(I) is a union of at least n MSI. Wituła [17] has generalized the description given in Theorem 13.1 onto the functions f : N → N preserving convergence of the real series (i.e. the functions for which the series ∑ a f (n) is convergent whenever the series ∑ an is convergent). For any subset A of N the notation: “the set A is a union of n (or at most n or at least n, respectively) MSI” means that there exists a family I of n (or at most n or at least n, respectively) mutually separated intervals of positive integers forming a partition of A. We say that the set I is a family of mutually separated intervals of positive integers if each element of I is an interval of N and if for any two distinct members K and L of I the following inequality min{|k − l| : k ∈ K ∧ l ∈ L} ≥ 2 holds. We will denote by P the family of all permutations of N. It is easy to check that Theorem 13.1 describes also all permutations p ∈ P rearranging any convergent series ∑ an with real terms into the convergent series ∑ a p(n) and preserving the sum of rearranged series, i.e. such that ∑ a p(n) = ∑ an . The family of all convergent permutations will be denoted by C. Notice that C is closed with respect to the composition of functions i.e. C is a semigroup (with the unity of course). The family of all divergent permutations will be denoted by D. Obviously we have D = P \ C. Moreover, we introduce the following notation CC, CD, DC

and

DD

for the nonempty subsets of P defined by the relation p ∈ AB

if and only if

p ∈ A and p−1 ∈ B

for any A, B ∈ {C, D} and p ∈ P. We present now the construction of some permutation p ∈ DC. Example 13.1. Let us put nk = k2 + 3k, for k ∈ N0 . Then nk+1 − nk = 2(k + 2), for k ∈ N. Next, let us set p(nk + i) = nk + 2i and p(nk + k + 2 + i) = nk + 2i − 1, for i = 1, 2, . . . , k + 2 and k ∈ N0 . Hence the set p([nk + 1, nk + k + 2])

13. New properties of the families of convergent and divergent permutations, I

195

is a union of (k + 2) MSI for every k ∈ N. Notice that for any interval I the set p−1 (I) is a union of at most 3 MSI. Therefore, by Theorems 13.1 and 13.2, we get p ∈ DC. Consequently, the families CD and DC are nonempty, as it was claimed above. In part II of this paper the following fundamental properties of all four "twosided" families of permutations, defined above, will be proven: U◦U = U and

(13.1) 

DD ◦ U = U ◦ DD = DD ∪ U =

D if U ⊂ D, D−1 otherwise,

(13.2)

where B−1 := {p−1 : p ∈ B} for every U = CD or DC and B ⊂ P, B 6= 0/ (in other words, both CD and DC are the subsemigroups of P)1 , and CC ◦ U = U ◦ CC = U

(13.3)

for every U = CC, CD, DC or DD (in fact, family CC is a proper subgroup of P). Family CC only seems to be small with respect to the composition of permutations. For example, it is proven in paper [16] that for any p ∈ D−1 and q ∈ D there exists a permutation ρ ∈ CC such that ρ 2 = idN and pρq ∈ DD. The symbol ◦ denotes here the composition of nonempty subsets of P defined by A ◦ B = {p ◦ q = pq : p ∈ A and q ∈ B} for every A, B ∈ P, A 6= 0, / B 6= 0/ and where pq(n) : = p(q(n)) for every n ∈ N. For brevity, we will write K < L for any two nonempty subsets K and L of N, whenever k < l for any k ∈ K and l ∈ L. In the sequel, we will write k < L (k > L, respectively) instead of {k} < L ({k} > L, respectively) for any k ∈ N and L ⊂ N, L 6= 0. / The symbol ⊂ denotes here the relation of being the proper subset, whereas the symbol ⊆ denotes the relation of being a subset.

1

It seems that, considering the subject-matter discussed here, within the framework of P = Sym(N) there appear some completely new problems of algebraic nature! Some classical results in this subject are connected with the results obtained by J. Schreier and S. Ulam, recalled, among others, in paper [7].

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Only the intervals of positive integers will be discussed here. For any n, m ∈ N, such that n ≤ m, we shall use the following notations: [n, m] := {n, n + 1, . . . , m}, [n, ∞) := {n, n + 1, . . .},

(n, ∞) := {n + 1, n + 2, . . .}, etc.

13.2 Group generated by the convergent permutations Algebraic and combinatoric properties of convergent and divergent permutations are discussed in many papers (see for example [2, 3, 11, 12, 13, 14]). Especially important is the paper by Pleasants [10] in which it is proven that the group G generated by C is not equal to P. We want to note here that the following description of the group G (see [22]) could be easily deduced: G = CD ∪ CD ◦ DC ∪ CD ◦ DC ◦ CD ∪ . . . = = DC ∪ DC ◦ CD ∪ DC ◦ CD ◦ DC ∪ . . . Let us set C1 := CD,

D1 := DC,

C2 := DC ◦ CD,

C3 := CD ◦ C2 ,

C4 := DC ◦ C3 ,

D2 := CD ◦ DC,

D3 := DC ◦ D2 ,

D4 := CD ◦ D3 ,

Then G =

S n∈N

Cn =

S

Dn and even G =

n∈N

S n∈N

C f (n) =

S

... ...

D f (n) for any increas-

n∈N

ing function f : N → N. We note that by (13.1) we get DC ◦ C2n = C2n ,

CD ◦ C2n−1 = C2n−1 ,

DC ◦ D2n−1 = D2n−1 ,

CD ◦ D2n = D2n ,

simultaneously we obtain Cn ◦ DC = Dn+1 , Cn ◦ CD = Cn ,

Dn ◦ CD = Cn+1 , Dn ◦ DC = Dn .

With the distinguished families Cn and Dn , n ∈ N, there is connected a number of interesting properties as well as several unsolved problems.

13. New properties of the families of convergent and divergent permutations, I

197

Theorem 13.3. The following inclusions hold P \ DD = C ∪ C−1 ⊂ C2 ∩ D2 , Cn ⊆ Cn+1 ,

Dn ⊆ Dn+1 ,

Cn ∪ Dn ⊆ Cn+1 ∩ Dn+1 .

(13.4) (13.5) (13.6)

Proof. (13.4): Let p ∈ CD and p ◦ p := q ∈ CD. Then we have p = q ◦ p−1 ∈ D2 , p = p−1 ◦ q ∈ C2 , i.e. p ∈ C2 ∩ D2 . Similar reasoning can be executed for permutation p ∈ DC. Thus CD ∪ DC ⊆ C2 ∩ D2 . Next, from equality (13.3) we get CC ⊂ C2 ∩ D2 . Inclusion (13.4) is sharp and the respective example is given in paper [16]. (13.5) and (13.6): We conduct the inductive proof. The case n = 1 obeys the inequality (13.4). Suppose that relations (13.5) and (13.6) hold for some n ∈ N. We have either Cn+2 = C2 ◦ Cn , if n is even, or Cn+1 = C2 ◦ Cn , if n is odd, i.e. D2 ◦ Cn = Cn+2 , which by (13.4) implies either Cn ∪ Cn+1 = (CD ∪ DC) ◦ Cn ⊆ C2 ◦ Cn = Cn+2 or Cn ∪ Cn+1 = (CD ∪ DC) ◦ Cn ⊆ D2 ◦ Cn = Cn+2 , i.e. Cn+1 ⊆ Cn+2 . Thus by (13.4) and by the inductive assumption we have Dn+2 = Dn ◦ D2 ⊇ Dn ◦ (CD ∪ DC) = Dn ∪ Cn+1 = Cn+1 , i.e. Cn+2 ∩ Dn+2 ⊇ Cn+1 .

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Roman Wituła, Edyta Hetmaniok and Damian Słota

Similarly we obtain the inclusions Dn+1 ⊆ Dn+2

and Cn+2 ∩ Dn+2 ⊇ Dn+1 ,

i.e. Cn+2 ∩ Dn+2 ⊇ Cn+1 ∪ Dn+1 . By virtue of the principle of mathematical induction it means that the inequalities (13.5) and (13.6) hold true for every n ∈ N. t u Problem 13.1. Are the inclusions (13.5) and (13.6) for n = 2, 3, . . . sharp? (The Authors suppose that they are. Moreover, the Authors think that Cn 6= Dn , n ∈ N.) Remark 13.1. The following result refers also to inclusions (13.4), (13.5) and (13.6) (we treat this result rather as a loyal supplement for our discussion – that is the answer to a problem: what if, in spite of everything, our supposition concerning the inclusions (13.5) and (13.6) and relation Cn 6= Dn , n ∈ N, is wrong.) Theorem 13.4. (see [19]) a) If An = An+1 for some A ∈ {C, D}, n ∈ N, then G = An . b) If for some n ∈ N either Cn ∪ Dn = Cn+1 or Cn ∪ Dn = Dn+1 , then we have G = Dn+1 = Cn+2

or G = Cn+1 = Dn+2 ,

respectively. c) If for some n ∈ N either Cn ⊆ Dn or Dn ⊆ Cn , then we have G = Dn+1 = Cn+2

or G = Cn+1 = Dn+2 ,

respectively. Sketch of the proof. b) Let n ∈ N be such that Cn ∪ Dn = Cn+1 . Then, with respect to (13.3), we obtain Dn+2 = Cn+1 ◦ DC = (Cn ◦ DC) ∪ (Dn ◦ DC) = Dn+1 ∪ Dn = Dn+1 , from which, in view of a), we get G = Dn+1 . Moreover, we have Cn+2 = DC ◦ Cn+1 = (DC ◦ Cn ) ∪ (DC ◦ Dn ) = Cn ∪ Dn+1 = Dn+1 ,

13. New properties of the families of convergent and divergent permutations, I

199

if only n ∈ 2N − 1, and Cn+2 = CD ◦ Cn+1 = (CD ◦ Cn ) ∪ (CD ◦ Dn ) = Cn ∪ Dn+1 = Dn+1 , if only n ∈ 2N. Thus, we have also Cn+2 = G. c) Let n ∈ N be such that Cn ⊆ Dn . Then we have Cn+1 = CD ◦ Cn ⊆ CD ◦ Dn = Dn , if n ∈ 2N, otherwise we have Cn+1 = DC ◦ Cn ⊆ DC ◦ Dn = Dn . So, we get Cn+1 = Dn and Cn ∪ Dn = Cn+1 which, by b), implies G = Cn+2 = Dn+1 .  Corollary 13.1. If G 6= Cn for every n ∈ N or equivalently G 6= Dn for every n ∈ N, then the following relations hold Cn \ Dn 6= 0, / Cn+1 \ (Cn ∪ Dn ) 6= 0/

and

Dn \ Cn 6= 0, / Dn+1 \ (Cn ∪ Dn ) 6= 0, /

for every n ∈ N. Problem 13.2. Is it true that if the equality holds Cn ∪ Dn = Cn+1 ∩ Dn+1 , for some n = n0 ∈ N, then it means that Cn = Cn+1

and

Dn = Dn+1 ,

for every n ≥ n0 ? Remark 13.2. Let S denote the family of all permutations p on N preserving the sum of series (i.e. satisfying the following condition: for every convergent series ∑ an of real terms, if the series ∑ a p(n) is also convergent then ∑ a p(n) = ∑ an ). The elements of family I := P \ S are called, after A. S. Kronrod [5], the substantially singular permutations p of N (singularity of this permutation consists in the existence of a convergent series ∑ an of real terms such that the series ∑ a p(n) is also convergent but the

200

Roman Wituła, Edyta Hetmaniok and Damian Słota

sums ∑ an and ∑ a p(n) are different). It is known that (see [23]): C2 ∩ I 6= 0/ 2 ,

D2 ⊂ S

and

D3 ∩ I 6= 0. /

Sketch of the proof. The inclusion D2 ⊂ S follows from the definition of D2 and the following characterization of sum preserving permutations (see Witula’s works [22, 21]): The permutation p ∈ P belongs to S iff there exists a natural number k = k(p) such that for each n ∈ N the nonempty finite sets An , Bn ⊂ N exist and satisfy the conditions: 1) p(An ) = Bn , 2) [1, n] ⊂ An , 3) each of sets An and Bn is a union of k MSI. The relation C2 ∩ I 6= 0/ results from the following relations (see [23]): F ⊆ C2

and F ∩ I 6= 0, /

where F denotes the family of all permutations p ∈ P, for which there exists the finite partition N1 , N2 , . . . , Nk of the set of natural numbers such that the restriction p|Ni is an increasing map for every i = 1, 2, . . . , k. The respective example of permutation p ∈ C2 ∩I is given also in paper [23]. Then the relation D3 ∩ I 6= 0/ results from the inclusion (13.6) for n = 2. t u Since I ⊂ DD and D2 ⊂ S, therefore by (13.1) and (13.2) we get (CD ∪ DC) ◦ I ⊂ DD.

(13.7)

For the contrast let us notice that (I ◦ DC) ∩ DC 6= 0/ since C2 ∩ I 6= 0. / From the combinatoric characterization of the permutations preserving the sum the following equality results CC ◦ S = S ◦ CC = S. In consequence, with respect to definition I = P \ S we also have CC ◦ I = I ◦ CC = I. Thus, the relation (13.7) takes the more general form (C ∪ C−1 ) ◦ I ⊂ DD. 2

(13.8)

We are troubled by the following problem: is the family C2 ∩ I = C2 \ S algebraically big (definition of the algebraically big subset of P is given in Section 4 of this paper)?

13. New properties of the families of convergent and divergent permutations, I

201

Historical remark Families C, D, CC, CD, DC, DD, C2 and D2 have been introduced for the first time by A. S. Kronrod [5] in 1946. He noticed that CC ◦ A = A ◦ CC = A

(13.9)

for each listed above subfamily A of family P. However, Kronrod did not discover any essential algebraic connections between these families of permutations. Additionally, let us notice that equality (13.9) holds also for A = Cn , Dn , n ∈ N, and for A = G.

13.3 Families D(k) We denote by D(k), for every k ∈ N, the set of all divergent permutations p for which there exists an increasing sequence {rn (p)}∞ n=1 of positive integers such that the set p−1 ([1, rn (p)]) is a union of at most k MSI for every n ∈ N. S D(k) ⊂ S (see Remark 13.2). Certainly k∈N

Remark 13.3. To prove that the permutation p does not belong to D(k), for given k ∈ N, we need to show that there exists N ∈ N such that the set p−1 ([1, n]) is a union of at least k + 1 MSI for every n ∈ N, n ≥ N. This fact will be used here in the proof of Theorem 13.6 and in Examples 13.2 and 13.3. Families D(k) are related, in a very interesting way, to the Riemann Derangement Theorem. For the fact, that for the given conditionally convergent series ∑ an and an interval I = [β , γ] ⊂ [−∞, ∞] there exists p ∈ D(k) such that the set of limit points of the p−rearranged series ∑ a p(n) is equal to I, it is necessary that i h γ−β γ+β γ−β γ+β ∑ an ∈ −(2k − 1) 2 + 2 , (2k − 1) 2 + 2 = [−(k − 1)γ + kβ , kγ − (k − 1)β ] (see [25, 24]). The basic algebraic properties of families D(k), k ∈ N, are given in the theorem presented below (the proof is omitted here).

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Theorem 13.5. The following relations hold DC ∩

S

D(k) = DC,

k∈N

DD \ D(k) 6= 0, / k∈N   S / DC ∩ D(k) \ D(l) 6= 0, l k and p(I) ≥ min p(T ) > (by (1)) > k, i.e. I > k and p(I) > k. So, I is the desired interval which terminates the proof. t u Remark 14.1. There exists a permutation p ∈ D such that, for any interval I, the set p(I) contains at most one interval J having the cardinality > 1 (see Example 1.3 in Part I). Remark 14.2. More subtle, than the one given in Lemma 14.1, combinatoric characterizations of the divergent permutations are given in papers [2] and [3].

14.2 The description of the families A ◦ B for A, B = CC, CD, DC or DD In this section three main theorems of this study will be presented.

14. New properties of the families of convergent and divergent permutations, II

215

Theorem 14.1. The product DC ◦ DC is equal to DC. Proof. Let p, q ∈ DC. Then p−1 , q−1 ∈ C

and (pq)−1 = q−1 p−1 ∈ C

because C is a semigroup. Suppose that pq ∈ C. Then also (pq)q−1 = p ∈ C, which is impossible. So pq ∈ DC and the inclusion below holds true DC ◦ DC ⊂ DC. Now let p ∈ DC. We show that there exist permutations p1 , p2 ∈ DC such that p2 p1 = p. Let us start with choosing two sequences Ik and Jk , k ∈ N, of intervals of N, satisfying, for every k ∈ N, the following assumptions: (1) Jk < Jk+1 , (2) Ik < p−1 (Jk ) < Ik+1 , (3) the set p(Ik ) is an union of at least k MSI, (4) there exists an interval Gk ⊂ p−1 (Jk ) such that the set p(Gk ) is an union of at least k MSI. Next we define the permutation p1 . Let p1 be an increasing map of the set S −1 S S −1 p (Jk ) onto the set Jk and let p1 (n) = p(n) outside the set p (Jk ). k∈N

k∈N

k∈N

Then the permutation p2 is given by p2 = pp−1 1 . By assumption (2), we have p−1 (Jk ) < p−1 (Jk+1 ), for every k ∈ N. Therefore, from assumptions (1) and (4) and from the definition of p1 , we see that the set p1 (Gk ) is an interval for every k ∈ N. Furthermore, from assumption (2) and from the definition of p1 , we have p1 (Ik ) = p(Ik ),

k ∈ N.

From this and from assumptions (3) and (4) we conclude that any of the following sets: p1 (Ik ) and

p2 p1 (Gk ) = p(Gk )

is a union of at least k MSI, for every k ∈ N. This shows that the permutations p1 and p2 are both divergent.

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Now, let s ∈ N be given so that the set p−1 (I) is a union of at most s MSI for any interval I. Then, by the definition of p1 and by the equality −1 p−1 1 (Jk ) = p (Jk ),

k ∈ N,

the set p−1 1 (I) is a union of at most s MSI, whenever I is a subinterval of Jk for some k ∈ N. Moreover, in view of the definition of p1 we have −1 p−1 1 (I) = p (I),

for each interval I such that ! either I ⊂

N\

[

Jk

or I = Jk for some k ∈ N.

k∈N

As the result we have that the set p−1 1 (I) is a union of at most 3s MSI, for every −1 interval I. Thus p1 ∈ C. To prove that p−1 2 ∈ C let us notice that   S (n) = n for every n ∈ N \ (i) p−1 J k , 2 k∈N

−1 (ii) the set p−1 2 (I) = p1 p (I) is a union of at most s MSI whenever I is a subinterval of Jk for some k ∈ N, (iii) p−1 2 (Jk ) = Jk for each k ∈ N.

Hence we easy deduce that p−1 2 (I) is a union of at most (2s + 1) MSI for every −1 interval I. So p2 ∈ C. The proof is completed. t u Corollary 14.1. We have CD◦CD = CD. More precisely, from the above proof it follows that for every p ∈ CD there exist permutations p1 , p2 ∈ CD such that p = p1 p2 , c(p2 ) ≤ 3 c(p) and c(p1 ) ≤ 1 + 2 c(p), where, for every convergent permutation q ∈ P, we set c(q) := sup{c(q; I) : I ⊂ N is an interval}, where c(q; A) := card(J), J is the family of MSI defined by the relation q(A) = S J for every A ⊂ N. Theorem 14.2. We have DC ◦ DD = DD ◦ DC = D and CD ◦ DD = DD ◦ CD = CD ∪ DD.

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217

Proof. First of all we note that if p ∈ DD and q ∈ DC then pq, qp ∈ D. Indeed, suppose that either pq ∈ C or qp ∈ C. Then p = (pq)q−1 = q−1 (qp) ∈ C ◦ C = C i.e. p ∈ C. This is a contradiction. So, both pq and qp are elements of D. In other words, the following conclusions hold: DC ◦ DD ⊂ D and D ⊃ DD ◦ DC and CD ◦ DD ⊂ CD ∪ DD and CD ∪ DD ⊃ DD ◦ CD. To prove the converse inclusions we consider four cases. First, suppose that p ∈ DD. We shall show that p = p2 p1 , for some permutations p1 ∈ DC and p2 ∈ DD. Suppose that the intervals Ik and Jk , k ∈ N, are chosen so that: (1) min I1 = 1, (2) 1 + max Ik = min Jk and 1 + max Jk = min Ik+1 , (3) cardJk = 2k, (4) there exist intervals Ek ⊂ Ik and Fk ⊂ p(Ik ) such that any of the two following sets: p(Ek ) and

p−1 (Fk )

is a union of at least k MSI. Let us put p1 (n) = n, for n ∈ k∈N Ik , and  2i + min Jk for i = 0, 1, ..., k − 1, p1 (i + min Jk ) = 2(i − k) + 1 + min Jk for i = k, k + 1, ..., 2k − 1, S

for k ∈ N, and let p2 = pp−1 1 . From this definition it results easily that p1 ∈ DC and that p2 p1 = p. Moreover, from conditions (2) and (3) we get that any of the two following sets: p2 (Ek ) = p(Ek ) and

−1 p−1 2 (Fk ) = p (Fk )

is a union of at least k MSI, for every k ∈ N. Hence, we have p2 ∈ DD as it was claimed. Let us set again that p ∈ DD. We will construct two permutations p1 ∈ DD and p2 ∈ DC such that p2 p1 = p. Assume that sequences Ik and Jk , k ∈

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N, of intervals obey the conditions (1)-(3) from above and, additionally, the following one: (5) for each k ∈ N, there exist intervals Gk ⊂ Ik

Hk ⊂ p−1 (Ik )

and

such that any of the two sets p(Hk ) and

p−1 (Gk )

is a union of at least k MSI. Let us set  p2 (i + min Jk ) =

2i + min Jk for i = 0, 2, ..., k − 1, 2(i − k) + 1 + min Jk for i = k, k + 1, ..., 2k − 1,

for k ∈ N and p2 (n) = n, for every n ∈ k∈N Ik . The permutation p1 is given by p2 p1 = p. The verification that p1 ∈ DD and p2 ∈ DC may be peformed in a similar way as previously and will be omitted here. Let us consider now the case p ∈ CD. We shall express p as the product p2 p1 of members p1 ∈ CD and p2 ∈ DD. We start by choosing the intervals In , Jn and Kn , n ∈ N, which form a partition of the set N and are such that S

(6) In < Jn < Kn < In+1 , (7) min p−1 (In ) < min p−1 (Jn ) < min p−1 (Kn ) and max p−1 (In ) < max p−1 (Jn ) < max p−1 (Kn ), (8) max p−1 (Jn ) < min p−1 (In+1 ) and max p−1 (Kn ) < max p−1 (Jn+1 ), (9) cardJn ≥ 2n, (10) moreover, there exist the subintervals Gn of In and Hn of Kn such that any of the following sets: p−1 (Gn ) and

p−1 (Hn )

is a union of at least n MSI and, the inclusion holds:  additionally, −1 −1 −1 (11) min p (Gn ), max p (Gn ) ⊂ p (In ), for every n ∈ N. Next, we define the permutations p1 and p2 .

14. New properties of the families of convergent and divergent permutations, II

219

Let us assume that p1 is an increasing map of the following sets: p−1 ({2i + min Jn : i = 0, 1, ..., n − 1}) , p−1 (Jn \ {2i + min In : i = 0, 1, ..., n − 1}) and p−1 (In ) onto the intervals [min Jn , n−1+min Jn ], [n+min Jn , max Jn ] and In , respectively, S −1 for every n ∈ N. Moreover, we set p1 (m) = p(m) for each m ∈ p (Kn ). n∈N

Since p1 ∈ P, we may define the permutation p2 by putting p2 (n) = pp−1 1 (n),

n ∈ N.

First we show that p1 ∈ CD. Let L be an interval. In view of the conditions (7) and (8) we may write L = L ∩ p−1 (I ∪ J ∪ K ∪ L) , where any of the following sets I and J is a union of at most three elements of the sequences {In } and {Jn }, respectively. The set K is a union of at most four elements of the sequence {Kn }, and L is an interval of N, which is a union of the successive elements of the sequence {In ∪ Jn ∪ Kn } such that p−1 (L) ⊂ L. Since p1 p−1 (U) = U, for any interval U = In , Jn or Kn , n ∈ N, then the set p1 (L) may be expressed in the form    (12) p1 (L) = L ∪ p1 L ∩ p−1 (I) ∪ p1 L ∩ p−1 (J) ∪ p1 L ∩ p−1 (K) . The following facts are the direct consequence of the definition of p1 . If  −1 U is an intervalthen the set p1 U ∩ p (In ) is a subinterval of In . The set p1 U ∩ p−1 (Jn ) is a union of at most two  subintervals of Jn and p1 U ∩ p−1 (Kn ) = p(U)  ∩ Kn −1 for every n ∈ N. Hence,  the set p1 L ∩ p (I) is a union of at most 3 MSI and the set p1 L ∩ p−1 (J) is a union of at most 6 MSI. On the other hand, if m ∈ N is chosen so that theset p(U) is a union of at most m MSI for any interval U, then p1 L ∩ p−1 (K) is a union of at most 4m MSI. Taking these observations together, by (12), we see that p1 (L) is a union of at most (4m + 10) MSI. So, −1 p1 ∈ C. By (10), each set p−1 1 (Hn ) = p (Hn ), n ∈ N, is a union of at least n −1 MSI and hence, p1 belongs to D. Therefore p1 ∈ CD as it was claimed.

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Now we have to show that p2 ∈ DD. Take a look at the following equality:

=

p2 ([min Jn , n − 1 + min Jn ])  −1 p2 p1 p ({2i + min Jn : i = 0, 1, ..., n − 1})

= {2i + min Jn : i = 0, 1, ..., n − 1} (by the definition of p1 ). We get that the set p2 ([min Jn , n − 1 + min Jn ]) is a union of n MSI, for every n ∈ N, and consequently p2 ∈ D. By using the conditions (10), (11) and the −1 definition of p1 we receive easily that the set p−1 2 (Gn ) = p1 p (Gn ) is a union −1 of at least n MSI. This implies that p2 ∈ D. Let us set again p ∈ CD. Now, our goal will be to construct the permutations p1 ∈ DD and p2 ∈ CD satisfying p2 p1 = p. Before we define p1 and p2 we need some basic assumptions. Let In and Jn , n ∈ N, be the increasing sequences of intervals such that the family {In : n ∈ N} ∪ {Jn : n ∈ N} forms the partition of N. Furthermore, we assume that the following conditions hold: (13) In < Jn < In+1 , (14) min p−1 (Jn ) < p−1 (In+1 ) < max p−1 (Jn+1 ), (15) p−1 (Jn ) < p−1 (Jn+1 ), (16) there is a subinterval Ωn of In such that the set p−1 (Ωn ) is a union of at least n MSI, (17) there exist four intervals: En , Gn ⊂ p−1 (Jn ) and Fn , Hn ⊂ Jn such that p−1 (Fn ) < En < p−1 (Hn ) < Gn , card(En ) = card(p−1 (Fn )) and

card(Gn ) = card(p−1 (Hn ))

and, additionally, any of the two following sets: p−1 (Fn ) and

p−1 (Hn )

is a union of at least n MSI, for every n ∈ N. It follows from (17) that p1 may be defined to be the increasing map of the following three sets:  En , p−1 (Hn ) and p−1 (Jn ) \ p−1 (Fn ) ∪ Gn onto the sets:

14. New properties of the families of convergent and divergent permutations, II

p−1 (Fn ),

and

Gn

221

 p−1 (Jn ) \ p−1 (Hn ) ∪ En ,

respectively, for every n ∈ N. Furthermore, we set p1 (i) = i

for any i ∈

[

p−1 (In ).

n∈N

Since p1 ∈P, then the permutation p2 is well defined by the equation p2 p1 = p. First, let us notice that, in view of the condition (17) and the definition of p1 , the permutation p1 belongs to DD. Next, since p2 (i) = p(i) for i ∈

[

p−1 (In )

n∈N

we receive, from (16), that p−1 2 ∈ D. We need only to show that p2 ∈ C. The proof of this fact is based on the following observations. If ∆ ⊆ N is an interval then we have   p2 ∆ ∩ p−1 (In ) = p ∆ ∩ p−1 (In ) = p(∆ ) ∩ In and if Γn := ∆ ∩ Gn then   −1 −1 −1 p2 (Γn ) = pp−1 1 (Γn ) = p p (Hn ) ∩ min p1 (Γn ), max p1 (Γn ) (by the definition of the restriction to p−1 (Hn ) of p1 )   −1 = Hn ∩ p min p−1 1 (Γn ), max p1 (Γn ) . Furthermore, if Φn := ∆ ∩ p−1 (Fn ) then, by the definition of the restriction to En of p1 , we get  −1  −1 p2 (Φn ) = pp−1 1 (Φn ) = p p1 (min Φn ), p1 (max Φn ) . Hence, if we choose m ∈ N in such a way that for every interval I the set p(I) is a union of at most m MSI then any of the following three sets:  p2 ∆ ∩ p−1 (In ) or p2 (Γn ) or p2 (Φn ) is a union of at most m MSI, for every n ∈ N. Let again ∆ be an interval of N. Then we have p2 ∆ ∩ p−1 (Jn ) \ p−1 (Hn ) ∪ En



= pp−1 ∆ ∩ p−1 (Jn ) \ p−1 (Hn ) ∪ En 1



= p ∆ ∗ ∩ p−1 (Jn ) \ p−1 (Fn ) ∪ Gn



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 (by the definition of the restriction to the set p−1 (Jn ) \ p−1 (Fn ) ∪ Gn of the permutation p1 , where ∆ ∗ is some interval of N)  = p p−1 (p(∆ ∗ ) ∩ Jn ) \ p−1 (Fn ) ∪ Gn = (p(∆ ∗ ) ∩ Jn ) \ (p(Gn ) ∪ Fn ) . The following set: (p(∆ ∗ ) ∩ Jn ) \ (p(Gn ) ∪ Fn ) is a union of at most (2m + 1) MSI, because the set p(∆ ∗ ) ∩ Jn is a union of at most m MSI and the set p(Gn ) ∪ Fn is a union  of at most (m + 1) MSI for −1 every n ∈ N. Therefore, the set p2 ∆ ∩ p (Jn ) is a union of at most (4m + 1) MSI. According to the conditions (14) and (15), any bounded interval ∆ may be written in the form ∆ = I ∪ J ∪ K, where the set K is a union of successive elements of the sequence {p−1 (In ∪ Jn ) : n ∈ N}, and the set I is an intersection of ∆ and at most four sets of the form p−1 (In ) satisfying the following relations: p−1 (In ) ∩ ∆ 6= 0/

and

p−1 (In ∪ Jn ) \ ∆ 6= 0. /

The set J is also an intersection of ∆ and at most three elements of the sequence p−1 (Jn ), n ∈ N, such that p−1 (Jn ) ∩ ∆ 6= 0/

and

p−1 (In ∪ Jn ) \ ∆ 6= 0. /

From the definition of p1 and the above considerations it follows that p2 (K) is an interval and that the set p2 (I) is a union of at most (4m) MSI and p2 (J) is a union of at most 3(4m + 1) MSI. Hence, the set p2 (∆ ) is a union of at most (16m + 4) MSI. Thus, p2 ∈ C as it was desired. t u Remark 14.3. Some parts of the above proof can be strengthen and, in consequence, the obtained conclusions can be stronger. For example, if we replace the condition (3) with (3’) cardJk = k t,

k ∈ N,

and we set p1 (i + s k + min Jk ) = it + s + min Jk ,

14. New properties of the families of convergent and divergent permutations, II

223

for every i = 0, 1, ..., k − 1, s = 0, 1, ...,t − 1, k ∈ N, then c∞ (p−1 1 ) = t, where for any q ∈ C we define c∞ (q) := lim max{c(q; I) : I ⊂ N is an interval such that I ≥ n}. n→∞

Consequently we receive the following result: For every p ∈ DD and t ∈ N, t ≥ 2, there exist permutations p1 ∈ DC, p2 , p3 ∈ DD, such that p = p2 p1 = p1 p3 and c∞ (p−1 1 ) = t. Theorem 14.3. We have U ◦ CC = CC ◦ U = U, for any U = CC, CD, DC or DD. Proof. In view of the equality C ◦ C = C and the fact that the identity permutation on N belongs to CC, it is easy to check that CC ◦ CC = CC

and CD ◦ CC = CC ◦ CD = CD.

Hence, we get DC ◦ CC = CC ◦ DC = DC

and

(CC ◦ DD) ∪ (DD ◦ CC) ⊂ D.

Now, if (CC◦DD)∩DC 6= 0/ then also DD∩(CC◦DC) 6= 0, / i.e. DD∩DC 6= 0, / which is impossible. So, CC ◦ DD = DD. Similarly, we show that DD ◦ CC = DD. t u

Acknowledgments The Author is greatly indebted to Dr J. Włodarz for his active interest in publication of this paper. The Author wants also to express sincere thanks to the Referee for many valuable remarks and advices enabling to correct the errors and to improve presentation of this work.

References [1] R. Wituła, E. Hetmaniok, D. Słota, Algebraic properties of the families of convergent and divergent permutations - Part I, ibidem.

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[2] R. Wituła, On the set of limit points of the partial sums of series rearranged by a given divergent permutation, J. Math. Anal. Appl. 362 (2010), 542-552. [3] R. Wituła, D. Słota, R. Seweryn, On Erd˝os’ Theorem for monotonic subsequences, Demonstratio Math. 40 (2007), 239-259.

ROMAN W ITUŁA Institute of Mathematics, Silesian University of Technology ul. Kaszubska 23, 44-100 Gliwice, Poland E-mail: roman.witula@polsl.pl