Families of Sets Without the Baire Property

Linköping Studies in Science and Technology. Dissertations No. 1825

Venuste NYAGAHAKWA

Department of Mathematics Division of Mathematics and Applied Mathematics Linköping University, SE–581 83 Linköping, Sweden Linköping 2017 Linköping Studies in Science and Technology. Dissertations No. 1825 Families of Sets Without the Baire Property

Venuste NYAGAHAKWA

[email protected] www.mai.liu.se Mathematics and Applied Mathematics Department of Mathematics Linköping University SE–581 83 Linköping Sweden

ISBN 978-91-7685-592-8 ISSN 0345-7524

Printed by LiU-Tryck, Linköping, Sweden 2017 To Rose, Roberto and Norberto

Abstract

The family of sets with the Baire property of a topological space X, i.e., sets which differ from open sets by meager sets, has different nice properties, like being closed under countable unions and differences. On the other hand, the family of sets without the Baire property of X is, in general, not closed under finite unions and intersections. This thesis focuses on the algebraic set-theoretic aspect of the families of sets without the Baire property which are not empty. It is composed of an introduction and five papers. In the first paper, we prove that the family of all subsets of R of the form (C \ M) ∪ N, where C is a finite union of Vitali sets and M,N are meager, is closed under finite unions. It consists of sets without the Baire property and it is invariant under translations of R. The results are extended to the space Rn for n ≥ 2 and to products of Rn with finite powers of the Sorgenfrey line. In the second paper, we suggest a way to build a countable decomposition ∞ {Xi}i=1 of a topological space X which has an open subset homeomorphic to (Rn, τ), n ≥ 1, where τ is some admissible extension of the Euclidean topology, ∞ such that the union of each non-empty proper subfamily of {Xi}i=1 does not have the Baire property in X. In the case when X is a separable metrizable ∞ manifold of finite dimension, each element of {Xi}i=1 can be chosen dense and zero-dimensional. In the third paper, we develop a theory of semigroups of sets with respect to the union of sets. The theory is applied to Vitali selectors of R to construct diverse abelian semigroups of sets without the Baire property. It is shown that in the family of such semigroups there is no element which contains all others. This leads to a supersemigroup of sets without the Baire property which contains all these semigroups and which is invariant under translations of R. All the considered semigroups are enlarged by the use of meager sets, and the construction is extended to Euclidean spaces Rn for n ≥ 2.

In the fourth paper, we consider the family V1(Q) of all finite unions of Vitali selectors of a topological group G having a countable dense subgroup Q. It is shown that the collection {G \ U : U ∈ V1(Q)} is a base for a topology τ(Q) on G. The space (G, τ(Q)) is T1, not Hausdorff and hyperconnected. It is proved that if Q1 and Q2 are countable dense subgroups of G such that Q1 ⊆ Q2 and the factor group Q2/Q1 is infinite (resp. finite) then τ(Q1) * τ(Q2) (resp. τ(Q1) ⊆ τ(Q2)). Nevertheless, we prove that all spaces constructed in this manner are homeomorphic. In the fifth paper, we investigate the relationship (inclusion or equality) between the families of sets with the Baire property for different topologies on the same underlying set. We also present some applications of the local function defined by the Euclidean topology on R and the ideal of meager sets there.

Populärvetenskaplig sammanfattning

Den familj av mängder som har Baire-egenskapen i ett topologiskt rum X, det vill säga mängder som är diﬀerensen mellan en öppen och en mager mängd, är som bekant en σ-algebra. Däremot är familjen av mängder som inte har Baire- egenskapen i X i allmänhet inte sluten under någon av de vanliga mängdoperatio- nerna. Denna avhandling, bestående av en inledning och fem artiklar, handlar om de algebraiskt mängdteoretiska aspekterna hos familjen av mängder som inte har Baire-egenskapen. I den första artikeln visar vi att familjen av delmängder av rummet R av formen (C\M)∪N, där C är en ändlig union av Vitali-mängder och M och N är magra, är sluten under ändliga unioner. Familjen består av mängder utan Baire-egenskapen och den är invariant under translationer. Vi utvidgar också dessa resultat till rum som är ändliga produkter av R och Sorgenfrey-linjen. I den andra artikeln konstruerar vi, för ett topologiskt rum X som har en öppen delmängd homeomorf med (Rn, τ), där τ är en tillåten utvidgning av den ∞ euklidiska topologin, en uppräknelig uppdelning {Xi}i=1 av X sådan att unionen av varje icketom äkta delfamilj inte har Baire-egenskapen i X. Om X är en ändlig- dimensionell, separabel och metriserbar mångfald så kan varje Xi väljas att vara tät och nolldimensionell. I den tredje artikeln utvecklar vi en teori för semigrupper av mängder, med union som gruppoperation, och tillämpar denna teori på Vitali-selektorer i R för att konstruera olika semigrupper av mängder som inte har Baire-egenskapen. Vi konstruerar sedan en semigrupp av mängder som innehåller alla dessa semigrupper och som består av mängder utan Baire-egenskapen och är invariant under translationer. Semigrupperna kan utvidgas med hjälp av magra mängder, och vi generaliserar resultaten till rummet Rn.

I den fjärde artikeln betraktar vi familjen V1(Q) bestående av ändliga unioner av Vitali-selektorer i en topologisk grupp G med en uppräknelig tät delgrupp Q. Vi visar att {G \ U : U ∈ V1(Q)} är en bas för en topologi τ(Q) på G. Rummet (G, τ(Q)) är T1 och hypersammanhängande men inte Hausdorﬀ. Vi visar också att om Q1 och Q2 är uppräkneliga täta delgrupper av G sådana att Q1 ⊆ Q2 och kvotgruppen Q2/Q1 är oändlig (respektive ändlig) så gäller att τ(Q1) * τ(Q2) (respektive τ(Q1) ⊆ τ(Q2)). Vi visar dock att alla topologiska rum konstruerade på detta sätt är homeomorfa. I den femte artikeln undersöker vi relationen mellan familjerna av mängder med Baire-egenskapen för olika topologier på samma underliggande mängd. Vi ger också några tillämpningar av den lokala funktionen hörande till den euklidiska topologin på R och idealet av magra mängder.

iii

Acknowledgments

This thesis appears in its current form due to the assistance, and guidance from several people. Therefore, I would like to express my sincere thanks to them. Foremost, I wish to express my gratitude to my main supervisor Vitalij A. Chatyrko for useful comments, remarks and encouragement throughout my graduate studies. Thank you Vitalij for helping me to improve my writing skills, and for your willingness to provide me advice about my future career. I also express my gratitude to my assistant supervisor Mats Aigner for his helpful collaboration. All insightful discussions we had together gave me additional energy for conducting research. I am also grateful to my second assistant supervisor Isidore Mahara whom I worked with every time I went back to Rwanda. In a very special way, I thank Bengt Ove Turesson, Björn Textorius, Martin Singull, Theresa Lagali Hensen, Meaza Abebe and Aksana Mushkavet for their invaluable assistance and support. I thank Tomas Sjödin, Theresia Carlsson Roth, Hans Lundmark and all members of the Department of Mathematics for building up and maintaining a high-quality working environment and for their help and support whenever needed at work. I am truly grateful to Froduald Minani and Alexandre Lyambabaje who not only taught me Mathematics but also encouraged and supported me in different kinds of academic activities. My gratitude also goes to the staff members of the UR-Sweden Programme for Research, Higher Education and Institutional Advancement, for their contribution to the smooth running of my studies. I am thankful to my fellow Ph.D students whom I shared so many excellent times at Linköping University. Thanks are also due to Jolanta Maria Pielaszkiewicz for having organized a wonderful trip to visit some historical sites in Poland. I am eternally grateful to my family and friends who supported and helped me along my studies by giving encouragement and providing the moral and emotional support when needed. I gratefully acknowledge the financial support received from the Swedish Inter- national Development Cooperation Agency funded through Linköping University and the University of Rwanda. Finally, I extend my sincere thanks to other people and institutions who supported me in one way or another. May the Lord God bless all of you.

Linköping, June 20, 2017 Venuste NYAGAHAKWA

List of papers

This thesis is based on the following appended papers and they are referred by their roman numerals.

I. V. A. Chatyrko and V. Nyagahakwa, On the families of sets without the Baire property generated by the Vitali sets, p-Adic Numbers, Ultrametric Analysis, and Applications, 2011, Vol.3, No.2, 100 - 107.

II. M. Aigner, V. A. Chatyrko and V. Nyagahakwa, On countable families of sets without the Baire property, Colloquium Mathematicum, 2013, Vol.133, No.2, 179 - 187.

III. M. Aigner, V. A. Chatyrko, and V. Nyagahakwa, The algebra of semigroups of sets, Mathematica Scandinavica, 2015, Vol.116, No.2, 161 - 170.

IV. V. A. Chatyrko and V. Nyagahakwa, Vitali selectors in topological groups and related semigroups of sets, Questions and Answers in General Topology, 2015, Vol.33, No.2, 93 - 102.

V. V. A. Chatyrko and V. Nyagahakwa, Sets with the Baire property in topologies formed from a given topology and ideals of sets, Accepted for publication in Questions and Answers in General Topology.

The research questions and results presented in the above papers were the product of multiple discussions between the authors. In Papers I and II, I was actively involved in discussions about the research subjects and the structure of the papers. For Paper III, all the three authors jointly contributed to writing

vii viii List of papers and deriving the results. In particular, I developed the part about the algebra of semigroups and ideals of sets. The results in Paper IV were developed jointly with my co-author. I wrote the part about diversity of semigroups of Vitali selectors. Regarding Paper V, I formulated the research topic and wrote the ﬁrst draft. The ﬁnal form of the paper was reached through discussions with my co-author.

Some parts of this thesis have been presented by me at the following conferences or seminars.

1. International Conference on Topology: On the Occasion of Filippo Cammaroto’s 65th Birthday, Messina-Italy, 7 - 11 September 2015.

2. The third International Modelling Week, University of Rwanda, College of Sci- ence and Technology, Kigali-Rwanda, 25 - 29 August 2014.

3. Analysis seminar & Mathematical Colloquium, Linköping University-Sweden, 2013, 2014, 2015, 2016.

4. The third EAUMP Conference: Advances in Mathematics and its Applications, Makerere University, Kampala-Uganda, 26 - 27 October 2016. Contents

I BACKGROUND AND SUMMARY 1

1 Introduction 3 1.1 Statement of the problem ...... 4 1.2 Aim of the thesis ...... 6

2 Background 9 2.1 Algebraic notions in Set Theory ...... 9 2.2 Sets with the Baire property ...... 11 2.3 Elements of topological groups ...... 12 2.4 Topological products of an inﬁnite family of spaces ...... 14 2.5 Extension of topologies via ideals ...... 14 2.6 Admissible extensions of topologies ...... 16 2.7 Vitali selectors on the real line ...... 17 2.8 Lebesgue covering dimension ...... 19

3 Summary of main results 21

Bibliography 27

ix x Contents

II INCLUDED PAPERS 29

I On the families of sets without the Baire property generated by the Vitali sets 31 1 Introduction ...... 33 2 Preliminary facts ...... 34 3 Main results ...... 35 4 Concluding remarks ...... 42 References ...... 42

II On countable families of sets without the Baire property 45 1 Introduction ...... 47 2 Auxiliary results ...... 48 3 A method of constructing countable families of sets without the Baire property ...... 51 4 Concluding remarks ...... 54 References ...... 55

IIIThe algebra of semigroups of sets 57 1 Introduction ...... 59 2 Auxiliary notions ...... 60 3 Semigroups of sets and ideals of sets ...... 60 4 Applications ...... 64 4.1 Two nested families of semigroups of sets ...... 64 4.2 Supersemigroups based on the Vitali sets ...... 65 4.3 A nonmeasurable case ...... 67 References ...... 67

IV Vitali selectors in topological groups and related semigroups of sets 69 1 Introduction ...... 71 2 Auxiliary notions ...... 73 2.1 Baire property ...... 73 2.2 Semigroups and ideals of sets ...... 73 2.3 A topology deﬁned by a semigroup ...... 74 xi

3 Vitali selectors in an abelian topological group G and related semigroup of sets ...... 74 3.1 A semigroup of Vitali selectors of G ...... 75 3.2 Diversity of semigroups of Vitali selectors ...... 76 3.3 Topologies on G deﬁned by semigroups of Vitali selectors . 78 References ...... 80

V Sets with the Baire property in topologies formed from a given topology and ideals of sets 83 1 Introduction ...... 85 2 Preliminary notions ...... 87 2.1 Ideal of sets and the Baire property ...... 87 2.2 Local functions and Kuratowski operators via ideals of sets, ∗-topologies ...... 88 2.3 Some natural relation between topologies and ideals . . . . 88 3 Two topologies on the same set and the Baire property ...... 89 3.1 π-compatible topologies ...... 89 3.2 A special case of π-compatible topologies ...... 90 3.3 Codense ideals of sets and sets with the Baire property . . 92 3.4 Partial equality of the families of nowhere dense sets (resp. meager sets or sets with the Baire property) ...... 95 4 Three and more topologies on the same set and the Baire property 96 4.1 Ideals of sets producing equal ∗-topologies ...... 96 4.2 Ideals of sets producing diﬀerent ∗-topologies ...... 98 5 Applications of the local function for proving some known facts . . 100 5.1 Applications to Bernstein sets ...... 100 5.2 Applications to Vitali sets ...... 101 References ...... 102

Part I

BACKGROUND AND SUMMARY

1 Introduction

In the early decades of the twentieth century, the work of French analysts E. Borel, R. Baire and H. Lebesgue provided the foundations of modern measure and integration theory as well as descriptive set theory. In their respective doctoral theses, they introduced the fundamental concepts related to measurability and category on the real line. Ever since, these concepts and their generalizations have been subject to intense investigation. In his thesis, "Sur quelques points de la théorie des fonctions", (1894), Borel created the first effective theory of measure [FK], extending the notion of length of intervals to a measure on a wide class of subsets of R. Borel considered measurable sets to be those subsets of R which are obtained by starting with intervals and forming the closure with respect to complementations and countable unions. The family of such sets is a σ-algebra and its elements are known as the Borel sets or the Borel-measurable sets on the real line. A few years later, Baire introduced basic concepts related to category and set up a classification of functions [GW]. In his thesis "Sur les fonctions de variables réelles", (1899), Baire defined nowhere dense sets, first category sets and second category sets on the real line, and proved the "Baire Category Theorem", a fundamental theorem in topology and functional analysis. Furthermore, he introduced the concept known today as the "Baire property". A set with the Baire property is a set which differs from an open set by a set of first category. The family of sets with the Baire property is a σ-algebra on R which contains all Borel sets as well as all first category sets on the real line. Inspired by Borel’s results, Lebesgue further extended the family of measur-

3 4 1 Introduction able sets on the real line [GW]. In his thesis, "Intégrale, longueur, aire", (1902), Lebesgue defined the outer measure of a subset E of R as the infimum, taken over all sequences of intervals covering E, of their total lengths. He also defined the inner measure of a set by using the outer measure and the operation of complement of sets. He then defined a set to be measurable if its outer measure equalled its inner measure, and the Lebesgue measure as their common value. In this way, he obtained a complete measure extending the Borel measure. The family of all Lebesgue measurable sets on R is a σ-algebra which contains all Borel-measurable subsets of the real line. In 1905, using the Axiom of Choice, the Italian mathematician G. Vitali constructed a first example of a subset of R which is neither Lebesgue measurable nor has the Baire property [Vi]. This set is known today as the Vitali set and its appearance has stimulated further developments in mathematics. Apart from Vitali sets, there exist other types of subsets of R (for example Bernstein sets and sets associated with Hamel bases) which are neither Lebesgue measurable nor have the Baire property [Ox, Kh1]. In fact the literature on such kind of sets is huge. We note that there exists a model of set theory in which every subset of the real line is Lebesgue measurable and has the Baire property. Indeed, in 1970, Solovay [So] showed that the existence of a non-Lebesgue measurable subset of the real line (or the existence of a subset of the real line without the Baire property) is unprovable if the Axiom of Choice is disallowed.

1.1 Statement of the problem

Consider the set R of real numbers endowed with the Euclidean topology and denote by P(R) the family of all subsets of R. Let Bp be the subfamily of P(R) consisting of sets with the Baire property in R and let L be the subfamily of P(R) consisting of Lebesgue measurable subsets of R. The families Bp and L have a number of similarities. Namely, they are σ-algebras of sets on R (and hence they are closed under all basic set operations) and they are invariant under the action of the group τ(R) of all translations of R, i.e., if A ∈ Bp (resp. A ∈ L) and h ∈ τ(R) then h(A) ∈ Bp (resp. h(A) ∈ L). As a σ-algebra, the family Bp is generated by the open sets of R together with the σ-ideal of all first category sets on R while the family L is generated by the open sets of R together with the σ-ideal of all subsets of R having the Lebesgue measure zero. Therefore, each of the families Bp and L contains the σ-algebra of all Borel sets of the real line. c As mentioned above, the complement Bp of Bp in P(R), as well as the complement Lc of L in P(R), is non-empty. In fact, each Vitali set, as well as each Bernstein set, does not have the Baire property on the real line and it is not measurable in the Lebesgue sense. Since each subset of R can be decomposed into a disjoint union of a first 1.1 Statement of the problem 5 category set and a set having the Lebesgue measure zero [Ox], none of the families Bp and L contains the other. More precisely, if we decompose a Vitali subset of R (or a Bernstein subset of R) into a disjoint union of a meager set and a null set, then one of the sets in this decomposition will be non-Lebesgue measurable but with the Baire property, and the other will be Lebesgue measurable but without the Baire property. c c It also follows that none of the families Bp and L contains each other. How- c c ever, unlike Bp and L, the families Bp and L are not closed under basic set operations. c c The intensive study of the families Bp and L was mostly concentrated on the construction of their elements with peculiar properties (cf. [Kh1]). For example, there are subgroups G of the additive group (R, +) of reals which are Bernstein sets c c (and hence such G belong to both families Bp and L ) and for which cardinality of the factor group (R, +)/G is continuum. However, this thesis was stimulated by the following question, posed by Chatyrko: c c Do there exist rich subfamilies of Bp (respectively, L ), which are invariant under the action of an infinite subgroup of the group H (R) of all homeomorphisms of R and on which we can define some algebraic structure from the set-theoretic point of view? One answer to this question was given by Chatyrko in [Ch1] where he showed that each finite union of Vitali sets does not possess the Baire property on the real line. He also observed that the family SV of all finite unions of Vitali sets is invariant under translations of R. From these facts, it follows that the family SV is an abelian semigroup of sets with respect to the operation of union of sets, that c it is invariant under translations of R, and that SV ⊆ Bp. ∗ Furthermore, it can be shown that the dual family SV = {R \ S : S ∈ SV } is also an abelian semigroup of sets with respect to the operation of intersection of ∗ c sets, that it is invariant under translations of R, and that SV ⊆ Bp. Similarly, in [Kh2] Kharazishvili showed that each finite union of Vitali sets is not measurable in the Lebesgue sense. Hence, we have also examples of semigroups ∗ of non-Lebesgue measurable sets (the same families SV and SV ) which constitute answers to the question posed above. c c Let us note that one can produce other subfamilies of Bp ∩ L which are invariant under translations of R and which are even abelian groups in some natural sense. For that, it is enough to consider the family F of all cosets (which are Bernstein sets) of a subgroup G mentioned above (the group operation on F is defined by a natural way).

Moreover, the family Ffin consisting of all finite unions of elements of F is an abelian semigroup of sets with respect to the operation of union of sets, it is c c invariant under translations of R and Ffin ⊆ Bp ∩ L . 6 1 Introduction

∗ Similarly, the dual family Ffin = {R\F : F ∈ Ffin} is also an abelian semigroup of sets with respect to the operation of intersection of sets, it is invariant under ∗ c c translations of R and Ffin ⊆ Bp ∩ L .

Let us point out some simple difference between elements of SV and elements of Ffin. Each element of Ffin as a Bernstein set must be everywhere dense in R however each element of SV does not need to be everywhere dense in R. c For more details and other results regarding elements of the families Bp, Bp, L and Lc we refer to [Ku, Ox, Kh1].

1.2 Aim of the thesis

The aim of this thesis is to present our contribution in the study of the families of sets with(out) the Baire property, and the family of non-measurable sets in the Lebesgue sense. Inspired by Chatyrko’s results [Ch1, Ch2] related to sets with(out) the Baire property on the real line, and Kharazishvili generalization of Vitali sets to Vitali selectors [Kh1], we obtain more extended results on the real line and even in more general topological spaces. Our contribution can be summarized into the following four points. Firstly, on the real line R we consider Vitali selectors related to different countable dense subgroups of the additive group (R, +). Following Chatyrko’s method and using subideals of meager sets on R, we produce diverse abelian semigroups with respect to the operation of union of sets. Those semigroups consist of sets without the Baire property, and they are invariant under translations of R. The dual semigroups of sets with respect to the operation of intersection of sets possess the same properties. We then extend these results to Euclidean spaces Rn, n ≥ 2, and to the products of Rn with finite powers of the Sorgenfrey line. Using the ideal of null sets, we also produce semigroups which consist of non-Lebesgue measurable sets, and which are invariant under translations of R (see Paper I and Paper III). Secondly, we study countable families of sets without the Baire property. We consider a topological space X, which has an open subset homeomorphic to the space (Rn, τ) for some admissible extension τ of the Euclidean topology on Rn, ∞ and suggest a way of constructing a countable decomposition {Xi}i=1 of X such ∞ that the union of each non-empty proper subfamily of {Xi}i=1 does not have the Baire property in X (see Paper II). Thirdly, we show that Vitali selectors related to an abelian Hausdorff nonmeager without isolated points topological group having countable dense subgroups can be used in the construction of a new topology on the underlying set. We then study properties of the new topological space, and show that any space obtained in this manner is hyperconnected and T1, and that any infinite product of such spaces with the box topology is hyperconnected, in particular, connected. Fur- 1.2 Aim of the thesis 7 thermore, we show that any two spaces obtained in this way are homeomorphic, but that no such a space can be a topological group (see Paper IV). Fourthly, we consider two topologies τ and σ on the same set X, and study conditions on τ and σ that imply a relationship (inclusion or equality) between the family of sets with the Baire property in (X, τ) and the family of sets with the Baire property in (X, σ). In particular, we consider the case in which τ and σ are ∗-topologies formed by using the local function and ideals of sets. Furthermore, we study the same problem for the families of nowhere dense sets and meager sets respectively (see Paper V). There are still many open problems to be further investigated. In this thesis we point out a few of them.

2 Background

The purpose of this chapter is to ﬁx notation and terminology, and to present some auxiliary facts from set theory, topological groups and dimension theory that are extensively used in our research papers.

2.1 Algebraic notions in Set Theory

In this section, we give a short introduction to families of sets with algebraic properties. By a family of sets we mean any set whose elements are themselves sets. Families of sets are mostly denoted by capital scripts letters like A , B, and so forth. Let X be a set and P(X) be the family of all subsets of X.

Deﬁnition 2.1. A family A ⊆ P(X) of sets is called a σ-algebra on X if the following conditions are satisﬁed:

(i) X ∈ A .

(ii) If A ∈ A then Ac ∈ A , where Ac = X \ A is the complement of A in X.

S∞ (iii) If A1,A2, ··· is a countable collection of sets in A then the union i=1 Ai is also in A .

From the De Morgan’s laws, each σ-algebra is also closed under countable intersections of sets and it contains also the empty set ∅.

9 10 2 Background

Deﬁnition 2.2. (a) A non-empty family I ⊆ P(X) of sets is called an ideal of sets on X if it satisﬁes the following conditions:

(i) If A ∈ I and B ∈ I then A ∪ B ∈ I . (ii) If A ∈ I and B ⊆ A then B ∈ I .

(b) If an ideal of sets I is closed under countable unions, then it is called a σ-ideal of sets on X.

Example 2.3. If A is a subset of X then the family IA of all subsets of A forms a σ-ideal of sets on X. The family If of all ﬁnite subsets of X forms an ideal of sets on X, but not a σ-ideal whenever X is inﬁnite. The family Ic of all countable subsets of X forms a σ-ideal of sets on X.

Given a family A of subsets of X, the smallest σ-algebra containing A is called the σ-algebra generated by A . It is the intersection of all σ-algebras containing the family A .

Example 2.4. Let (R, τE) be the real line, i.e., the set of real numbers R endowed with the topology τE deﬁned by all open intervals of R. The σ-algebra generated by the collection of all open intervals of R is called the Borel σ-algebra, and its elements are called the Borel sets. The σ-algebra generated by the ideal of Lebesgue measure zero sets (null sets) together with the Borel sets equals the collection L of all Lebesgue measurable subsets of R.

Another type of family of sets which is important in this work is a semigroup of sets. Deﬁnition 2.5. A non-empty family S ⊆ P(X) of sets is called a semigroup of sets with respect to the operation of union (resp. intersection) of sets on X if S is closed under ﬁnite unions (resp. intersections) of sets, i.e., if A1,A2, ··· ,An ∈ S Sn Tn then i=1 Ai ∈ S (resp. i=1 Ai ∈ S).

Note that if S is a semigroup of sets with respect to the operation of union of sets then the dual family S∗ = {X \ S : S ∈ S} is also a semigroup of sets with respect to the operation of intersection of sets, and vice versa. In this thesis we will discuss only semigroups of sets with respect to the operation of union of sets (shortly, semigroups of sets).

Example 2.6. Let A be a non-empty family of subsets of X. Deﬁne SA = Sn { i=1 Ai : Ai ∈ A , n ∈ N}, where N is the set of positive integers. It is clear that SA is a semigroup of sets on X. We will call SA the semigroup of sets generated by A .

We now present two ways of extending a given semigroup by the use of an ideal of sets. Let A and B be families of subsets of X. Deﬁne A ∨ B = {A ∪ B : A ∈ A ,B ∈ B} and A ∗ B = {(A \ B1) ∪ B2 : A ∈ A ,B1 ∈ B,B2 ∈ B}. 2.2 Sets with the Baire property 11

Proposition 2.7. Let S be a semigroup of sets and let I be an ideal of sets on X. Then the families S ∨ I and S ∗ I are semigroups of sets on X such that S ⊆ S ∨ I ⊆ S ∗ I . Moreover, (S ∗ I ) ∗ I = S ∗ I and (S ∨ I ) ∨ I = S ∨ I .

The proof is elementary and it is based on the deﬁnition of ideals of sets.

2.2 Sets with the Baire property

In this section, we recall some deﬁnitions and basic facts related to sets with the Baire property. This property is a classical topological notion which is originated in the thesis of R. Baire. Below X is assumed to be a non-empty topological space. For a subset A of X, the notation ClX (A) (resp. IntX (A)) stands for the closure (resp. interior) of A in X. Deﬁnition 2.8. Let A be a subset of X. The set A is said to be a nowhere dense set in X if its closure has an empty interior, i.e., IntX ClX (A) = ∅.

The set A is said to be dense in X if the closure of A is X, i.e., ClX (A) = X.

It can be shown that every subset of a nowhere dense set is also a nowhere dense set, and that any ﬁnite union of nowhere dense sets is again a nowhere dense set. Hence, the family of all nowhere dense sets in X forms an ideal of sets. Note that if A is dense in X and A ⊆ B ⊆ X then B is also dense in X. Example 2.9. Let R be the real line. The set Z of all integers is a nowhere dense subset of R. However, the set Q of rational numbers, as well as its extensions Q(α) = {a + bα : a ∈ Q, b ∈ Q} for each irrational number α, is a (countable) dense subset of R.

It is not true in general that a countable union of nowhere dense sets is a nowhere dense set. In fact, the set Q is a union of countably many nowhere dense sets in R, however IntR ClR(Q) = R. Deﬁnition 2.10. A subset A of X is called meager (or of ﬁrst category) in X if A can be represented as a countable union of nowhere dense sets in X. Any set that is not meager in X is called nonmeager (or of second category) in X.

The family of all meager subsets of X forms a σ-ideal of sets. The fundamental theorem of R. Baire asserts the following [KF]. Theorem 2.11 (Baire Category Theorem). Let X be a complete metric space. Then X cannot be covered by countably many nowhere dense subsets of X. More- over, the union of countably many nowhere dense subsets of X has a dense complement. 12 2 Background

Recall that the real line R is a complete metric space. Consequently, by the Baire Category Theorem, R is of second category. Note that meager sets (resp. sets with Lebesgue measure zero) on R are small in a topological point of view (resp. in a measure theoretical point of view). Nevertheless, a set that is small in one sense may be large in some other sense.

Theorem 2.12 ([Ox]). The real line R can be decomposed into two complementary sets A and B such that A is of ﬁrst category and B is of Lebesgue measure zero.

It follows from Theorem 2.12 that every subset of the real line can be represented as a disjoint union of a null set and a set of ﬁrst category.

Deﬁnition 2.13. A subset A of X is said to have the Baire property in X if it can be represented in the form A = (O \ M) ∪ N, where O is an open set of X and M,N are meager sets in X.

Note that a subset A of X has the Baire property in X iﬀ there are an open set O of X and a meager set M of X such that A = O∆M, where ∆ is the symmetric diﬀerence set operation.

The family Bp(X) of sets with the Baire property in X is a σ-algebra of sets, which is generated by the family M of all meager sets in X together with the family O of all open sets in X. Furthermore, it is invariant under the action of the group H (X) of all homeomophisms of X, i.e., if A ∈ Bp(X) and h ∈ H (X) then h(A) ∈ Bp(X). Let us also notice that the family O is a semigroup of sets on X such that SO = O, and that Bp(X) = O ∗ M. An important question is to know whether every subset of X possesses the Baire property in X. There exist topologies on X such that the family of sets having the Baire property in X contains every subset of X, whereas there exist topologies on X such that the family of sets with the Baire property in X does not contain every subset of X. In Section 2.7, we will present examples of sets without the Baire property on the real line. For a more detailed description of sets with the Baire property, we refer to [Ku, Ox].

2.3 Elements of topological groups

The results which are presented in the fourth paper partially concern topological groups. Thereupon, in this section, we present some notions and concepts which are related to topological groups.

Deﬁnition 2.14. A set G which is endowed with the structures of a(n abelian) group and a topological space is called a(n abelian) topological group if the multiplicative mapping G × G −→ G, (x, y) 7−→ x · y, and the inverse mapping 2.3 Elements of topological groups 13

G −→ G, x 7−→ x−1 are continuous, where G × G is viewed as a topological space by using the product topology.

Note that we have used a multiplicative group notation for the group operation. We give some examples of abelian topological groups. The real numbers R with the addition x + y : x, y ∈ R as the group operation endowed with the standard topology, the non-zero real numbers R \{0} with the product x · y : x, y ∈ R as the group operation endowed with the subspace topology from the standard topology on R, the set T of complex numbers of modulus one with the product z1 · z2 : z1, z2 ∈ T as the group operation endowed with the subspace topology from the complex plane, ﬁnite products Rn, Tn and the products Rn ×Tm, n ≥ 1, m ≥ 1, where the group operations and topologies are deﬁned naturally. Recall that a topological space X is called:

a T0-space if for any two distinct points of X, there is an open set which contains one point but not the other;

a T1-space if each point of X is a closed set; a Hausdorﬀ space if for any two distinct points p, q of X there are disjoint open sets U, V such that p ∈ U and q ∈ V ;

a regular T1-space if X is T1 and for any point p and any closed set F such that p∈ / F there are disjoint open sets U, V such that p ∈ U and F ⊆ V ;

a Tychonoff space if X is a T1-space and for each closed set F ⊆ X and each point x ∈ X \ F , there exists a continuous function h : X −→ [0, 1] such that h(F ) = 0 and h(x) = 1; a normal Hausdorff space if X is Hausdorff and for any disjoint closed sets F and G there exist disjoint open sets U, V such that F ⊆ U and G ⊆ V .

Let us recall that each normal Hausdorff (resp. Tychonoff, regular T1, Haus- dorff or T1) space is Tychonoff (resp. regular T1, Hausdorff, T1 or T0). The existence of a compatible group structure on a topological space guarantees good topological properties of the space.

Proposition 2.15 ([HR]). Let G be a topological group. If G is a T0-space then G is Tychonoﬀ.

Let us note that not every topological group is normal Hausdorﬀ. For instance, the group Zm, where m is any uncountable cardinal number, is a Tychonoﬀ space but it is not normal [HR]. However, if our group is additionally locally compact the situation will be changed.

Proposition 2.16 ([HR]). Every locally compact T0-topological group is normal Hausdorﬀ.

Let us note that not every topology on a group makes it into a topological group. For example, let G be an arbitrary infinite group endowed with the topol- 14 2 Background ogy τ = {U ⊆ G : U = ∅ or G \ U is finite}. It can be shown that τ is the weakest possible topology on G which makes G into a T1-space. However, G is not a topological group. In fact, G satisfies the T0-separation axiom but G is not Hausdorff. For other notions and facts, we refer the reader to [HR, Mo].

2.4 Topological products of an inﬁnite family of spaces

Let Xα, α ∈ I, be a nondegenerated topological space for each α ∈ I and I inﬁnite. Then Q the product topology on the product α∈I Xα, is that topology which has a Q basis for its open sets the collection of all sets of the form α∈I Uα, where each Uα is open in Xα, and Uα 6= Xα for only ﬁnitely many α; Q the box product topology on the product α∈I Xα is that topology which has Q a basis for its open sets the collection of all sets of the form α∈I Uα, where Uα is open in Xα for each α ∈ I.

It is well known that if each topological space Xα, α ∈ I, is connected then Q α∈I Xα with the product topology is also connected as a topological space. Q However, α∈I Xα with the box product topology is disconnected as a topological space whenever each Xα is regular T1. If a similar statement holds for Hausdorﬀ spaces it is unknown. Q Let us note that there are examples of α∈I Xα with the box product topology which are connected. In the fourth paper we present such products where all spaces are T1.

2.5 Extension of topologies via ideals

In this section, we describe a method which is used to extend topologies by using ideals of sets. The method was introduced by Janković and Hamlett in their joint work [JH]. Many results which are presented in the ﬁfth paper are based on topologies which are constructed by the use of this method. Let (X, τ) be a topological space, I an ideal of sets on X and let A be a subset of X. For each element x of X denote by N (x) = {U ∈ τ : x ∈ U} the family of all open neighborhoods at the point x.

Deﬁnition 2.17 ([Ha]). The set A∗(I ) = {x ∈ X : A ∩ U/∈ I for every U ∈ N (x)} is called the local function of A with respect to the ideal of sets I and the topology τ. 2.5 Extension of topologies via ideals 15

The following theorem establishes some basic properties of the local function.

Theorem 2.18 ([JH]). Let (X, τ) be a topological space, I and J be ideals of sets on X and let A and B be subsets of X. The following statements hold:

(i) If A ⊆ B then A∗(I ) ⊆ B∗(I ). (iv) [A∗(I )]∗(I ) ⊆ A∗(I ).

(v) If I ∈ I then (A ∪ I)∗(I ) = (ii) (A ∪ B)∗(I ) = A∗(I ) ∪ B∗(I ). (A \ I)∗(I ) = A∗(I ).

(iii) A∗(I ) = Cl(A∗(I )) ⊆ Cl(A). (vi) If I ⊆ J then A∗(J ) ⊆ A∗(I ).

Recall that a point x ∈ X is said to be an ω-accumulation (resp. a condensation) point of A ⊆ X if every neighborhood of x contains an inﬁnite (resp. uncountable) number of points of A.

Example 2.19. We present some examples of local functions: If I = {∅} then ∗ ∗ ∗ A (I ) = Cl(A). If I = P(X) then A (I ) = ∅. If I = If then A (I ) is the ∗ set of all ω-accumulation points of A. If I = Ic then A (I ) is the set of all condensation points of A. If A is an element of I then A∗(I ) = ∅ for every ideal of sets I on X.

It is well known [Ku] that if Cl is a Kuratowski closure operator, i.e., Cl is an operator from P(X) to P(X) such that Cl(∅) = ∅,A ⊆ Cl(A), Cl(A ∪ B) = Cl(A) ∪ Cl(B) and Cl(Cl(A)) = Cl(A) for all A, B ⊆ X, then there exists a unique topology on X whose closed sets are given by the collection {F ⊆ X : Cl(F ) = F }. By Theorem 2.18, the operator Cl∗(.) from P(X) to P(X) deﬁned by Cl∗(A) = A ∪ A∗(I ) for every subset A of X, is a Kuratowski closure operator. Hence the family {U ⊆ X : Cl∗(X \ U) = X \ U} is a topology on X, which is denoted by τ ∗(I ). The topology τ ∗(I ) is called the ∗-topology (or new topology) from the old topology τ and the ideal of sets I . It has the collection β(τ, I ) = {O \ I : O ∈ τ, I ∈ I } as a base. Hence τ ⊆ τ ∗(I ). One of the basic properties of the ∗-topologies is that each element I ∈ I is closed and discrete in (X, τ ∗(I )). By Example 2.19, it follows that if I = {∅} then Cl∗(A) = Cl(A), and hence τ ∗(I ) = τ. Furthermore, if I = P(X) then Cl∗(A) = A, and hence τ ∗(I ) = τdisc, where τdisc is the discrete topology on X.

It follows from the Theorem 2.18 (vi) that if I1 and I2 are ideals of sets on ∗ X such that I1 ⊆ I2 and A ⊆ X is closed in τ (I1) then A is also closed in ∗ ∗ ∗ ∗ τ (I2), i.e., τ (I1) ⊆ τ (I2). Hence, τ ⊆ τ (I ) ⊆ τdisc for every ideal I of sets on X. Moreover, there exist ideals of sets I diﬀerent from I = {∅} such that τ ∗(I ) = τ and there exist ideals I of sets on X diﬀerent from P(X) such that ∗ τ (I ) = τdisc. For other notions and facts, we refer the reader to [Ku, JH]. 16 2 Background

2.6 Admissible extensions of topologies

Given two diﬀerent topologies τ and σ on the same underlying set X, one can be interested in the study of conditions on τ and σ that imply some relationship (for example inclusion or equality) between the family of sets with the Baire property in (X, τ) and the family of sets with the Baire property in (X, σ).

Deﬁnition 2.20 ([CN]). Let τ and σ be topologies on a set X. The topology σ is said to be an admissible extension of the topology τ on X if

(i) τ ⊆ σ, and

(ii) τ is a π-base of σ, i.e., for each non-empty element O of σ there is a non- empty element V in τ such that V ⊆ O.

Now we explain why the concept of admissible extension is important.

Theorem 2.21. Let X be a set, τ and σ be topologies on X such that σ is an admissible extension of τ. Then the spaces (X, τ) and (X, σ) have the same families of nowhere dense sets (resp. meager sets or sets with the Baire property).

Proof. Let N be a nowhere dense set in the space (X, τ). We will show that N is also a nowhere dense set in the space (X, σ). Consider a non-empty set Oσ ∈ σ. Since σ is an admissible extension of τ, there exists a non-empty set Oτ ∈ τ such that Oτ ⊆ Oσ. Since N is a nowhere dense set in the space (X, τ), there is a non-empty set Vτ ∈ τ such that Vτ ⊆ Oτ and Vτ ∩ N = ∅. Note that Vτ ∈ σ and Vτ ⊆ Oσ. Hence we get that N is a nowhere dense set in the space (X, σ). The inverse implication can be proved by a similar simple argument. The case of meager sets is obvious. Since τ ⊆ σ, it is clear that each subset A of X with the Baire property in (X, τ) also has the Baire property in (X, σ). We refer to [ACN] for the other inclusion.

Example 2.22 (Sorgenfrey line). The ﬁrst example of an admissible extension of the Euclidean topology τE is the Sorgenfrey topology τS on R. Recall that the base of τS is given by the collection {[a, b): a, b ∈ R, a < b} and the topological space (R, τS) is called the Sorgenfrey line. In general, for each integer k ≥ 1 and k m k+m k Qk each integer m ≥ 0 the topology τS × τE on R , where τS = j=1(τS)j and k+m k+m (τS)j = τS for each j ≤ k, is an admissible extension of τE on R .

More general examples of admissible extensions of the Euclidean topology τE on R can be found in [CH].

Example 2.23 (Hattori spaces). Let A be a subset of R. In his work [Ht], Hattori deﬁned a topology τ(A) on R in the following way: 2.7 Vitali selectors on the real line 17

(i) for each x ∈ A, {(x − ε, x + ε): ε > 0} is the neighborhood base at x.

(ii) for each x ∈ R \ A, {[x, x + ε): ε > 0} is the neighborhood base at x.

In their joint work [CH], Chatyrko and Hattori continued to study the topological spaces (R, τ(A)),A ⊆ R. They observed that for any subsets A, B of R, A ⊇ B iﬀ τ(A) ⊆ τ(B), in particular, τ(R) = τE ⊆ τ(A) and τ(B) ⊆ τ(∅) = τS. Furthermore, they showed that for every A ⊆ R the topology τ(A) is an admissible extension of τE on R. Hence the collection {τ(A): A ⊆ R} is a poset of topologies on R, which consists of admissible extensions of the topology τE on R. The topological spaces (R, τ(A)),A ⊆ R are known as Hattori spaces. All the spaces (R, τ(A)),A ⊆ R are regular hereditarily Lindelöf and hereditarily separable.

Other examples of admissible extensions of the topology τE on R, for which the construction is diﬀerent from the previous ones, can be obtained by using the family of nowhere dense sets on the real line.

Example 2.24 (Admissible extensions via nowhere dense sets). Consider a subideal I of the family of nowhere dense sets of the real line. The topology ∗ ∗ τE(I ) is an admissible extension of τE on R. In fact, since τE ⊆ τE(I ), it is ∗ enough to show that τE is a π-base of τE(I ). Let O be a non-empty element of ∗ τE(I ). We can assume that O = U \ I where ∅= 6 U ∈ τE and I ∈ I . Since I is a nowhere dense set in (R, τE), there exists a non-empty element V of τE such that V ⊆ U and V ∩ I = ∅. It is clear that V ⊆ O.

Note that each topology τ(A),A (R from the Example 2.23 cannot be a ∗- topology from the Euclidean topology on the reals. Hence the topologies described in Example 2.23 are diﬀerent from those described in the Example 2.24.

2.7 Vitali selectors on the real line

In this section, we present two types of sets which cannot be Lebesgue measurable and which cannot have the Baire property on the real line. The ﬁrst type of such sets is called Vitali selectors. The Vitali selectors are closely related to Vitali sets which were introduced by G. Vitali in 1905. Let Q be a countable dense subgroup of the additive group (R, +). For an element x ∈ R, denote by Tx the translation of R by x, i.e., Tx(y) = y + x for each element y ∈ R. If A is a subset of R and x ∈ R, denote the set Tx(A) = {a + x : a ∈ A} by Ax. Deﬁne the equivalence relation E on R as follows: for x, y ∈ R, let xEy if and only if x − y ∈ Q, and let {Eα(Q): α ∈ I} be the set of all equivalence classes, where I is some indexing set. Observe that the cardinality |I| of I is equal to c (the continuum), and that for each α ∈ I and each x ∈ Eα(Q) we have Eα(Q) = Q + x. Therefore, each equivalence class Eα(Q) is dense in R. 18 2 Background

Deﬁnition 2.25 ([Kh1]). A Vitali Q-selector of R is any subset V of R such that |V ∩ Eα(Q)| = 1 for each α ∈ I. A Vitali Q-selector is called a Vitali set [Vi] whenever the subgroup Q coincides with the group Q of rational numbers.

For Vitali selectors, the following statements hold.

Proposition 2.26. Let V be a Vitali Q-selector of R. Then we have the following.

(i) If q1, q2 ∈ Q and q1 6= q2 then (V + q1) ∩ (V + q2) = ∅. S (ii) R = q∈Q(V + q).

(iii) The set V is not meager in R.

(iv) The set V + x is also a Vitali Q-selector for each x ∈ R.

Let us point out that there exist Vitali Q-selectors of the real line which have some additional properties. In fact, there exist bounded Vitali Q-selectors and there exist Vitali Q-selectors which are dense in any non-empty open subsets of R (see [Ch2], Propositions 3.1 and 3.3).

Theorem 2.27 ([Kh1]). Any Vitali Q-selector of R is non-Lebesgue measurable and does not have the Baire property on the real line.

Let us note that Theorem 2.27 is a particular case of the following more general result; which is valid also for Vitali selectors of the real line.

Theorem 2.28 ([Ch1]). Let Vi be a Vitali set for each i ≤ n where n is some Sn integer such that n ≥ 1. Then the set U = i=1 Vi does not contain the diﬀerence O \ M, where O is a non-empty open set and M is meager. In particular, the set U does not possess the Baire property on the real line.

A similar result to Theorem 2.28 about non-Lebesgue measurability of ﬁnite unions of Vitali sets was proved by Kharazishvili. For a proof, we refer to [Kh2]. Let us notice that in the light of Proposition 2.26 (ii), the results of Theorem 2.28 cannot be extended to all countable union of Vitali sets. However, Chatyrko proved the following statement which is valid not only for Vitali sets but also for Vitali selectors of R.

Theorem 2.29 ([Ch2]). If V is a Vitali set and Γ is a non-empty proper subset S of Q then the set q∈Γ(V + q) does not possess the Baire property on the real line.

The second type of sets which neither possess the Baire property on the real line, nor are measurable in the Lebesgue sense, is called Bernstein sets. Bernstein sets were constructed by F. Bernstein in 1908 and they are deﬁned as follows. 2.8 Lebesgue covering dimension 19

Deﬁnition 2.30 (cf. [Ox]). A subset B of R is called a Bernstein set if F ∩B 6= ∅ and F ∩ (R \ B) 6= ∅ for every uncountable closed subset F of R.

This deﬁnition shows directly that the complement R \ B of a Bernstein set B in R is also a Bernstein set. The construction of Bernstein sets is based on the method of transﬁnite induction. For the construction of Bernstein sets, we refer to [Ox, Kh1].

Theorem 2.31 (cf. [Ox]). Any Bernstein set B is non-measurable in the Lebesgue sense and does not have the Baire property. Indeed, every Lebesgue measurable subset of either B or R \ B is a null set, and any subset of B or R \ B that has the Baire property is of ﬁrst category.

Remark 2.32. Apart from Vitali selectors and Bernstein sets, there are other types of sets which are neither Lebesgue measurable, nor have the Baire property on the real line. For instance, non-Lebesgue measurable sets (or sets without the Baire property) associated with Hamel bases. Moreover (see for instance [Kh1]), there exist non-Lebesgue measurable sets (or sets without the Baire property) which have algebraic structure of subgroups of the additive group (R, +).

In a similar way, we can construct Vitali selectors in higher-dimensional Eu- clidean spaces Rn. However, for n ≥ 2, there exists a subclass of Vitali selectors that we called rectangular Vitali selectors in [N]. These are Vitali selectors which are related to countable dense subgroups of (Rn, +) that are assumed to be product of n countable dense subgroups of (R, +). Hence, each rectangular Vitali selector of Rn is a product of n Vitali selectors of R.

2.8 Lebesgue covering dimension

In this section, we review some basic properties of the Lebesgue covering dimension dim in separable metrizable spaces.

Let X be a topological space and let A = {Aα}α∈Γ be a family of subsets, not all empty, of X, where Γ is some indexing set.

The order of the family A = {Aα}α∈Γ is the largest integer n for which there T exists a subset I of Γ with n + 1 elements such that α∈I Aα is non-empty, or ∞ if there is no such largest integer. S The family A = {Aα}α∈Γ is said to be a cover of X if α∈Γ Aα = X. A cover B is a reﬁnement of another cover A of the same space X (in other words B reﬁnes A ), if for every B ∈ B there exists an A ∈ A such that B ⊆ A.

Deﬁnition 2.33. Let X be a topological space. Then

• dim X = −1 if and only if X = ∅. 20 2 Background

• dim X ≤ n, where n = −1, 0, 1, ··· , if every finite open cover of the space X has a finite open refinement of order not exceeding n.

• dim X = n if it is true that dim X ≤ n but it is not true that dim X ≤ n − 1.

• dim X = ∞ if for every integer n it is false that dim X ≤ n for n = −1, 0, 1, ··· .

If dim X = n, then X is called an n-dimensional topological space. Let us remind that a topological space X is said to be separable if it contains a countable dense subset. The space X is said to be metrizable if there exists a metric on X which induces the topology on X. For separable metrizable spaces, some basic properties of the Lebesgue covering dimension are summarized in the following theorems [En].

Theorem 2.34 (Fundamental Theorem of Dimension). For every natural number n, we have dim Rn = n.

Theorem 2.35 (Subspace Theorem). If Y is a subspace of a separable metrizable space X, then dim Y ≤ dim X.

Theorem 2.36 (Countable Sum Theorem). Let X be a separable metrizable space S∞ and X = i=1 Fi, where Fi is closed in X for each i. If dim Fi ≤ n for each i, then dim X ≤ n.

Theorem 2.37 (Decomposition Theorem). A separable metrizable space X sat- isﬁes the inequality dim X ≤ n if and only if X can be represented as the union of n + 1 subspaces Z1,Z2, ··· ,Zn+1 such that dim Zi ≤ 0 for i = 1, 2, ··· , n + 1.

Theorem 2.38 (Brouwer Dimension Theorem). Let Y be a subset of Rn, n ∈ N. n Then dim Y = n if and only if IntR (Y ) 6= ∅. Theorem 2.39 (Product Theorem). For every pair X,Y of separable metrizable spaces, not both empty, we have dim(X × Y ) ≤ dim X + dim Y .

Apart from the Lebesgue covering dimension, there exist other dimension functions; for instance the small inductive dimension ind and the large inductive dimension Ind. The three dimensions coincide in the class of separable metrizable spaces, i.e., ind X = Ind X = dim X for every separable metrizable space X. However, in general spaces, the three functions ind, Ind and dim do not need to coincide. We refer to [En] for other facts related to the three dimension functions. 3 Summary of main results

Paper I: On the families of sets without the Baire property generated by the Vitali sets

It is well known that the family of sets with the Baire property on the real line is invariant under translations of R, and that it is closed under all basic set operations. On the other hand, the family of sets without the Baire property is also invariant under translations of R but it is not closed under basic set operations. One can pose the following question: Do there exist rich families of subsets of R without the Baire property, which are invariant under the action of an inﬁnite subgroup of the group H (R) of all homeomorphisms of R, and on which we can deﬁne some algebraic structure from the set-theoretic point of view? In this paper, we present a positive answer to this question, not only on the real line, but also in higher-dimensional Euclidean spaces Rn, n ≥ 2. Furthermore, similar results are derived in the case where Rn is endowed with some admissible extensions of the Euclidean topology.

On the real line R, we define the family V2 = {(U \M)∪N : U ∈ V1; M,N ∈ A}, where V1 is the family of all finite unions of all Vitali sets of R, and A is the σ-ideal of meager subsets of R. We prove that the family V2 consists of zero-dimensional sets without the Baire property, that it is invariant under translations of R, and that it is an abelian semigroup of sets with respect to the operation of union of sets. For the Euclidean spaces Rn, n ≥ 2, we define a Vitali set of Rn as a product Qn i=1 Vi of Vitali sets Vi, i = 1, 2, ··· , n of the real line R (see [N] for a general

21 22 3 Summary of main results

n n n case of Vitali sets), and put V2 = {(U \ M) ∪ N : U ∈ V1 ; M,N ∈ A }, where n n n V1 is the family of all ﬁnite union of all Vitali sets of R , and A is the σ-ideal of n n all meager sets on R . We prove that the family V2 consists of sets without the Baire property in Rn, that it is invariant under translations of Rn, and that it is an abelian semigroup of sets with respect to the operation of union of sets. We n also prove that 0 ≤ dim A ≤ n − 1 for each element A of V2 .

Finally, we introduce a notion of an admissible extension of a topology τ1 on a set X as a new larger than τ1 topology τ2 with some peculiar property on the same set X. With the help of the notion we extend the mentioned above results from the Euclidean spaces Rn to products of Rn with ﬁnite powers of the Sorgenfrey line.

Paper II: On countable families of sets without the Baire property

It is well known that any Vitali set does not possess the Baire property on the real line. The same result is valid for finite unions of Vitali sets. It was proved by Chatyrko [Ch2] that if V is a Vitali set and Γ is a non-empty proper subset S of the set Q of rational numbers, then the set q∈Γ(V + q) does not possess the Baire property on the real line. He also extended the result to higher-dimensional Euclidean spaces Rn for n ≥ 2. In this paper, we generalize Chatyrko’s result by replacing Vitali sets by Vitali selectors, and by changing the Euclidean spaces to the spaces (Rn, τ) for some admissible extension τ of the Euclidean topology on Rn, where n is some positive integer. Let Q be a countable dense subgroup of the additive group (Rn, +). We prove that if V is a Vitali Q-selector of Rn (see the description of Vitali sets in Rn, Paper S I) and Γ is any non-empty proper subset of Q then the set q∈Γ(V + q) does not possess the Baire property in (Rn, τ), where τ is any admissible extension of the Euclidean topology on Rn. Furthermore, let X be a topological space which has an open subset homeomorphic to the space (Rn, τ) for some admissible extension τ of the Euclidean topology on Rn. We prove that there exists an infinite disjoint countable family ∞ S∞ {Xi}i=1 of sets in X such that i=1 Xi = X, and such that the union of each ∞ non-empty proper subfamily of {Xi}i=1 does not have the Baire property in X. In case X is a separable metrizable finite-dimensional manifold, we prove that each ∞ element of the family {Xi}i=1 can be chosen dense and zero-dimensional.

Paper III: The Algebra of semigroups of sets

In the first paper, we constructed an abelian semigroup consisting of sets without the Baire property on the real line. In this paper, we continue the study by looking for other families which consist of sets without the Baire property, which 23 are invariant under translations of R, and which are semigroups of sets with respect to the operation of union of sets. We develop a theory of semigroups with respect to the operation of union of sets and ideals of sets. By applying this theory to Vitali selectors of R, we obtain diverse abelian semigroups which consist of sets without the Baire property, and which are invariant under translations of R. In particular, many results from the first paper can be proven within this theory. So first, we consider any countable dense subgroup Q of the additive group (R, +), and the family V1(Q) of all finite unions of Vitali Q-selectors of R (ana- logues of Vitali sets of R). We note that the family V1(Q) is an abelian semigroup which consists of sets without the Baire property in R, and that it is invariant under translations of R.

Next, we consider the collection {V1(Q): Q is a countable dense subgroup of (R, +)} of those semigroups, and study the relationship (in the sense of inclusion) between members of this collection. In particular, we prove that there is no semi- sup group V1(Q) which contains all others. This leads us to deﬁne a semigroup V1 based on all Vitali selectors which we call a supersemigroup of Vitali selectors. sup Naturally, the supersemigroup V1 contains all semigroups V1(Q), where Q is sup any countable dense subgroup of (R, +). We prove that the supersemigroup V1 consists of sets without the Baire property in R, and that it is invariant under translations of R. Finally, all the above mentioned semigroups are enlarged with the help of meager sets on R to semigroups whose elements are of the form (U \ M) ∪ N, where U ∈ V1(Q) and M,N are meager subsets of R. We then generalize the construction to ﬁnite-dimensional Euclidean spaces Rn for n ≥ 2. Note that a more extended description can be found in [N].

Paper IV: Vitali selectors in topological groups and related semigroups of sets

Let G be an abelian Hausdorﬀ nonmeager without isolated points topological group having countable dense subgroups. In this paper, we use Vitali selectors of G to deﬁne a new topology on G, and then study its topological properties.

Let Q be a countable dense subgroup of G, and let V1(Q) be the family of all finite unions of Vitali selectors of G related to the subgroup Q. We prove that the collection {G \ U : U ∈ V1(Q)} is a base for a topology τ(Q) on G, which makes G into a new topological space that is denoted by G(Q). Furthermore, we study the topological properties of the space G(Q). We prove that G(Q) is a T1-space but it is not Hausdorff (and hence it cannot be a topological group), and that it is hyperconnected. As a consequence, any infinite product of the spaces G(Q), where Q is a countable dense subgroup of G, in the box topology, is also hyperconnected, in particular, connected. This is a step to answer a question 24 3 Summary of main results

posed by Chatyrko and Karassev [CK]: Let Xα, α ∈ A, be an infinite system of nondegenerated connected Hausdorff spaces. Under what conditions on the system Qb is the space α∈A Xα (dis)connected? We also study the relationship (inclusion or equality) between the topologies τ(Q1) and τ(Q2), when Q1 and Q2 are countable dense subgroups of G such that Q1 ⊆ Q2. We proved that if the factor group Q2/Q1 is finite then τ(Q1) ⊆ τ(Q2), and that if the factor group Q2/Q1 is infinite then τ(Q1) * τ(Q2). Moreover, we prove that for any given countable dense subgroups Q1 and Q2 of G the topological spaces G(Q1) and G(Q2) are homeomorphic.

Paper V: Sets with the Baire property in topologies formed from a given topology and ideals of sets

Let X be a non-empty set and let τ1, τ2 be topologies on X. It is well known that the family of sets with the Baire property in (X, τi) contains all open sets and all meager sets of (X, τi), i = 1, 2. However, depending on the topology τi, the family of sets with the Baire property in (X, τi) can contain or not contain every subset of X. This implies that neither the family of sets with the Baire property in (X, τ1) nor the family of sets with the Baire property in (X, τ2) needs to contain the other. One can pose the following question: What relationship between the topologies τi, i = 1, 2 implies a relationship (inclusion or equality) between the families of sets with the Baire property in (X, τi), i = 1, 2? In this paper, we focus on answering this question, with a particular attention to the case in which the topology τ2 is formed with the help of a local function deﬁned by the topology τ1 and an ideal of sets on X. We ﬁrst introduce a concept generalizing the notion of admissible extension, that we call the π-compatibility. Two topologies τ1 and τ2 on X are said to be π-compatible if for each non-empty element O ∈ τ2 there is a non-empty element V ∈ τ1 which is a subset of O, and vice versa. We prove that if τ1 and τ2 are π-compatible then the spaces (X, τ1) and (X, τ2) have the same families of sets with the Baire property.

We next consider the case where τ2 is deﬁned from τ1 and an ideal of sets I ∗ ∗ on X, that is, τ2 = τ1 (I ). For this case, we study conditions under which τ1 (I ) is an admissible extension of τ1. We prove that if I is a subideal of nowhere ∗ dense sets in (X, τ1) then τ1 (I ) is an admissible extension of τ1 on X, and hence ∗ the spaces (X, τ1) and (X, τ1 (I )) have the same families of sets with the Baire property. In addition, we prove that if (X, τ1) is a second countable topological space and I is a σ-subideal of meager sets in (X, τ1) such that τ1 ∩ I = {∅} then ∗ the spaces (X, τ1) and (X, τ1 (I )) have the same families of sets with the Baire property. 25

Several other conditions implying a relationship between sets with the Baire ∗ property in (X, τ1) and sets with the Baire property in (X, τ1 (I )) are derived. We also provide conditions under which diﬀerent ideals of sets produce the same topologies on X. We note that a similar question can be posed for any other families of sets obtained by using the topologies, and the opposite question can be also posed. In this paper, we also consider the question for nowhere dense sets and meager sets. However, we observe that the results for sets with the Baire property do not always carry over nicely to nowhere dense sets and to meager sets.

Bibliography

[ACN] M. Aigner, V. A. Chatyrko and V. Nyagahakwa, On countable families of sets without the Baire property, Colloquium Mathematicum, 2013, Vol.133, No.2, 179 - 187.

[Ch1] V. A. Chatyrko, On Vitali sets and their unions, Matematicki Vesnik, 2011, Vol.63, No.2, 87 - 92.

[Ch2] V. A. Chatyrko, On countable unions of nonmeager sets in hereditarily Lindelöf spaces, p-Adic Numbers, Ultrametric Analysis, and Applications, 2011, Vol.3, No.1, 1 - 6.

[CH] V. A. Chatyrko and Y. Hattori, A poset of topologies on the set of real numbers, Comment. Math. Univ. Carolin., 2013, Vol.54, No.2, 189 - 196.

[CN] V. A. Chatyrko and V. Nyagahakwa, On the families of sets without the Baire property generated by the Vitali sets, p-Adic Numbers, Ultrametric Analysis, and Applications, 2011, Vol.3, No.2, 100 - 107.

[CK] V. A. Chatyrko and A. Karassev, The (dis)connectedness of products in the box topology, Questions and Answers in General Topology, 2013, Vol.31, No.1, 1 – 21.

[En] R. Engelking, Theory of dimensions, ﬁnite and inﬁnite, Sigma series in pure mathematics, Volume 10, Heldermann Verlag, 1995.

[FK] M. Foreman and A. Kanamori, Handbook of set theory, Volume 1, Springer Science + Business Media B.V., 2010.

27 28 Bibliography

[GW] D. M. Gabbay, J.H. Woods and A. Kanamori, Handbook of the history of logic: Sets and extensions in the twentieth century, Vol.6, Elsevier, 2012.

[Ha] E. Hayashi, Topologies deﬁned by local properties, Math. Ann., 1964, Vol.156, No.3, 205 - 215.

[HR] E. Hewitt, K.A Ross, Abstract harmonic analysis: Structure of topological groups, Integration theory, Group representations, Volume I, Springer-Verlag Berlin Heidelberg, 1963.

[Ht] Y. Hattori, Order and topological structures of poset of the formal balls on metric spaces, Mem.Fac.Sci.Eng. Shimane Univ.Ser. B: Math. Sci., 2010, Vol. 43, 13-26.

[JH] D. Janković and T.R. Hamlett, New topologies from old via Ideals, Amer. Math. Monthly, 1990, Vol.97, No.4, 295 - 310.

[KF] A. N Kolmogorov and S.V Fomin, Introductory real analysis, Courier Dove Publications, 1975.

[Kh1] A.B. Kharazishvili, Nonmeasurable sets and functions, Vol.195, Elsevier, 2004.

[Kh2] A.B Kharazishvili, Measurability properties of Vitali sets, The American Mathematical Monthly, 2011, Vol.118, No.8, 693 - 703.

[Ku] K. Kuratowski, Topology, Volume I, PWN, Warszawa and Academic Press, New York, 1966.

[Mo] S. A. Morris, Pontryagin duality and the structure of locally compact abelian groups, Cambridge University Press, 1977.

[N] V. Nyagahakwa, Semigroups of sets without the Baire property in ﬁnite dimensional Euclidean spaces, Licentiate thesis, Linköping University, 2015.

[Ox] J.C. Oxtoby, Measure and Category: Graduate texts in mathematics, Springer-Verlag, New York, Heldenberg, Berlin, 1971.

[So] R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. Math., 1970, Vol. 92, No.1, 1 - 56.

[Vi] G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, Italy, 1905. Part II

INCLUDED PAPERS

Paper I On the families of sets without the Baire property generated by the Vitali sets

Authors: Vitalij A. Chatyrko and Venuste Nyagahakwa

This paper has been published as: Vitalij A. Chatyrko and Venuste Nyagahakwa, On the families of sets without the Baire property generated by the Vitali sets, p-Adic Numbers, Ultrametric Analysis, and Applications, 2011, Vol.3, No.2, 100 - 107.

On the families of sets without the Baire property generated by the Vitali sets

Vitalij A. Chatyrko∗ and Venuste Nyagahakwa∗†

Abstract

Let A be the family of all meager sets of the real line R, V be the family of all Vitali sets of R, V1 be the family of all ﬁnite unions of elements of V and V2 = {(C \A1)∪A2 : C ∈ V1; A1,A2 ∈ A}. We show that the families V, V1, V2 are invariant under translations of R, and V1, V2 are abelian semigroups with respect to the operation of unions of sets. Moreover, V ⊂ V1 ⊂ V2 and V2 consists of zero-dimensional sets without the Baire property. Then we extend the results above to the Euclidean spaces Rn, n ≥ 2, and their products with the ﬁnite powers of the Sorgenfrey line.

Keywords: Vitali set of Rn, Baire property, Nonmeasurable in the sense of Lebesgue, Sorgenfrey line.

1 Introduction

It is well known that the family A of meager sets on the real line R is a σ-ideal of sets, i.e., A contains all subsets of its members and all countable unions of its members. An interesting extension of A in the family 2R of all subsets of R is the family Bp of sets possessing the Baire property; the σ-algebra of sets generated by the σ-ideal A and the σ-algebra B of Borel sets. c R R Recall that the complement Bp = 2 \ Bp in 2 is not empty (for example, c each Vitali set S of R is an element of Bp) and the considered classes A, Bp, B c and Bp are invariant under the translations of R. Moreover, some elements of c Bp can have algebraic structures (see [Kh] for subgroups of the additive group

∗Department of Mathematics, Linköping University, SE–581 83 Linköping, Sweden, E-mails: [email protected], [email protected]. †Department of Mathematics, University of Rwanda, P.O. Box 3900 Kigali, Rwanda, E-mail: [email protected].

33 34Paper I On the families of sets without the Baire property generated by the Vitali sets

c R, which are the elements of Bp, and other interesting examples and references). This paper was stimulated by the following simple question. c Is there in the family Bp a suﬃciently rich, invariant under translations of R, subfamily F on which we can deﬁne some algebraic structure? So instead of traditionally to focus our attention on algebraic structures of some c c elements of Bp we look for subfamilies of Bp having some algebraic structures.

Let V be the family of all Vitali sets of R, V1 be the family of all ﬁnite unions of Vitali sets, V2 = {(C \ A1) ∪ A2 : C ∈ V1,A1,A2 ∈ A} and Z be the family of all zero-dimensional subsets of R. We have the following statement. c Theorem 1.1. (i) V ⊂ V1 ⊂ V2 ⊂ Bp ∩ Z;

(ii) The families V, V1, V2 are invariant under translations of R;

(iii) The families V1, V2 are abelian semigroups with respect to the operation of unions of sets.

Remark 1.2. There exists a countable subfamily µ of V consisting of all trans- S c S lations of some ﬁxed Vitali set such that µ ∈ Bp but dim( µ) = 1 (see Ex- S S∞ ample 3.7). Note that the set µ is the union i=1 µi of an increasing sequence µ1 ⊂ µ2 ⊂ · · · of elements of V1 (so dim µi = 0 for each i).

Then we extend the results above to the Euclidean spaces Rn, n ≥ 2, and their products with the finite powers of the Sorgenfrey line. Let us recall that the Sorgenfrey line is the set of real numbers R with the topology τS defined by the half-open intervals [a, b), a, b ∈ R, and dim(R, τS) = 0, where dim is the covering dimension for the Tychonoff spaces from [En].

2 Preliminary facts

We will recall some known notions and statements (see, for example, [Ox] and [En]). Let X be a topological space. A subset A of X is called meager in X if A is the union of countably many nowhere dense sets in X, otherwise A is called nonmeager in X. A subset A of X is said to possess the Baire property in X iﬀ there is an open set O and two meager sets M,N such that A = (O \ M) ∪ N.

Let R be the real line. For x ∈ R denote by Tx the translation of R by x, i.e., Tx(y) = y + x for each y ∈ R. If A is a subset of R and x ∈ R, we denote Tx(A) by Ax. The equivalence relation E on R is deﬁned as follows. For x, y ∈ R, let xEy iﬀ x − y ∈ Q, where Q is the set of rational numbers. Let us note its equivalence classes by Eα, α ∈ I. Note that |I| = c (continuum). It is evident that for each α ∈ I and each x ∈ Eα,Eα = Qx. Hence every equivalence class Eα is dense in R. 3 Main results 35

A Vitali set of R here is any subset S of R such that |S ∩ Eα| = 1 for each

α ∈ I [Vi]. Note that if r1, r2 ∈ Q and r1 6= r2 then Sr1 ∩ Sr2 = ∅ and [ Sr = R (1) r∈Q Each Vitali set does not possess the Baire property in R.

3 Main results

Let X be a set and V, O and M be families of subsets of X satisfying the following properties:

(a) the family V consists of non-empty elements;

(b) for each ﬁnite union U of elements of V (i.e. U is the union of ﬁnitely many elements of V) and each non-empty element O of O there is an element V of V such that V ⊂ O \ U;

(d) each element M of M contains no element of V.

For the families V, O and M we have the following statement. Proposition 3.1. Let U be a ﬁnite union of elements of V and E,F ∈ M. Then

(i) the set U ∪ E contains no non-empty element O of O; (ii) the set U ∪ E contains no diﬀerence O \ M, where O is a non-empty element of O and M is an element of M;

(iii) the set (U \ F ) ∪ E cannot be written as the union (O \ M) ∪ N, where O ∈ O and M,N ∈ M.

Proof. (i) Assume that U ∪ E ⊃ O for some non-empty O ∈ O. By (b) there is V ∈ V such that V ⊂ O \ U. Note that V ⊂ (U ∪ E) \ U ⊂ E. But this is a contradiction with (d). (ii) Assume that U ∪ E ⊃ O \ M for some non-empty O ∈ O and some M ∈ M. Note that U ∪ (E ∪ M) = (U ∪ E) ∪ M ⊃ (O \ M) ∪ M ⊃ O. By (c) we have E ∪ M ∈ M what leads to a contradiction with (i). (iii) Assume that (U \F )∪E = (O \M)∪N for some O ∈ O and some M,N ∈ M. Note that the set O must be non-empty. Really, if O = ∅ then (U \F )∪E = N and so U ⊂ N ∪ F . Note that by (c) we have N ∪ F ∈ M what leads to a contradiction with (d). Hence, O 6= ∅. Note further that U ∪ E ⊃ (U \ F ) ∪ E ⊃ O \ M what leads to a contradiction with (ii). 36Paper I On the families of sets without the Baire property generated by the Vitali sets

Lemma 3.2. The family V of all Vitali sets of the real line R as V and the family O of all open sets of R as O satisfy the conditions (a) and (b).

Sn Proof. Let U = i=1 S(i), where S(i) is a Vitali set of R for each i ≤ n, and O be a non-empty open set of R. Note that Eα ∩ (O \ U) 6= ∅ for each α ∈ I. So we can choose a Vitali set S such that S ⊂ O and S ∩ U = ∅. Hence S ⊂ O \ U.

Let M0 be the family of sets of Lebesgue measure zero of the real line R. It is well known that M0 is a σ-ideal of sets. A natural extension of M0 in 2R is the family M of all measurable (in the sense of Lebesgue) sets; the σ-algebra of c c sets generated by M0 and B. It is well known that V ⊂ M , where M is the complement of M in 2R.

Proposition 3.3. Let U ∈ V1 and E,F be meager sets (resp. sets of Lebesgue measure zero) of R. Then

(i) the set U ∪ E contains no difference O \ M, where O is a non-empty open set and M is a meager set (resp. any set of Lebesgue measure zero) (note (see [Ox]) that for every non-empty open set O there are some meager set A and some set B of Lebesgue measure zero such that O = A ∪ B; so the set U ∪ A contains the difference O \ B and the set U ∪ B contains the difference O \ A);

(ii) the set (U \ F ) ∪ E is zero-dimensional and it cannot be written as the union (O \ M) ∪ N, where O is an open set and M,N are meager sets (resp. any sets of Lebesgue measure zero).

Proof. Since the families A (resp. M0) and V considered as M and V satisfy the conditions (c) and (d), the statements (i) and (ii) follow from Proposition 3.1 and Lemma 3.2. We need to give only one comment to (ii). Since (U \ F ) ∪ E ⊂ U ∪ E, it follows from (i) that the set (U \ F ) ∪ E does not contain any non-empty open set. Recall that each subset A of the real line R with dim A = 1 must contain a non-empty open interval [En]. Hence, the set (U \ F ) ∪ E is zero-dimensional.

Put V3 = {(C \ A1) ∪ A2 : C ∈ V1; A1 ∈ M0,A2 ∈ M0}.

Lemma 3.4. Let V1,V2 ∈ V2 (resp. V3). Then V1 ∪ V2 ∈ V2 (resp. V3).

Proof. Straightforward.

Lemma 3.5. For each Vitali set S and each x ∈ R, the set Sx is also a Vitali set.

Proof. The statement follows from the following observations: 3 Main results 37

(1) Since for any diﬀerent elements s1, s2 of S we have (x+s1)−(x+s2) = s1 −s2, the numbers x + s1 and x + s2 belong to the diﬀerent equivalence classes Eα, α ∈ I.

(2) Consider α ∈ I and let sα ∈ S ∩ Eα. Note that there is β ∈ I such that sα − x ∈ Eβ. Moreover, there are sβ ∈ Eβ ∩ S and r ∈ Q such that sα − x = sβ + r. So x + sβ = sα − r ∈ Eα.

Proof of Theorem 1.1. (i) The inclusions V ⊂ V1 ⊂ V2 are obvious, the c inclusion V2 ⊂ Bp ∩ Z follows from Proposition 3.3 (ii). (ii) The statement is valid because the meager sets are invariant under translations of R and due to the Lemma 3.5. (iii) It follows from the Lemma 3.4 that the family V2 is an abelian semigroup with respect to the operation of unions of sets. The statement about the family V1 is obvious. Let us recall the following: 0 Theorem 3.6 ([Ch]). If S is a Vitali set of R and Q is a non-empty proper subset S 0 c of Q then {Sr : r ∈ Q } ∈ Bp. Example 3.7. Let us consider a Vitali set S of R such that S ⊂ (−1, 1). Put 0 S 0 Q = Q ∩ (−2, 2) and U = {Sr : r ∈ Q }. It follows from the Equality (1) that c U ⊃ (−1, 1). So dim U = 1. Note that by Theorem 3.6 we have U ∈ Bp.

n n n n n Consider now the Euclidean space R for some n > 1. Let A (resp. Bp , B , M0 and Mn) be the family of all meager sets (resp. sets with the Baire property, Borel sets, sets of Lebesgue measure zero and measurable sets in the sense of Lebesgue) of Rn. n n For each vector x ∈ R denote Tx the translation of R by x, i.e., Tx(y) = y +x n n n for each y ∈ R . If A is a subset of R and x ∈ R we denote Tx(A) by Ax. It is Qn evident that if A is the product j=1 Aj of Aj ⊂ R, j ≤ n and x = (x1, ··· , xn) Qn n n n n n then Ax = j=1(Aj)xj . Note that the families A , Bp , B , B0 and M are invariant under translations of Rn. n Qn A Vitali set of R here is any set S = j=1 S(j), where S(j) ∈ V for each n j = 1, 2, ··· , n. Note that if r1, r2 ∈ Q and r1 6= r2 then Sr1 ∩ Sr2 = ∅ and [ n Sr = R (2) n r∈Q Let us note that each Vitali set of Rn is Lebesgue nonmeasurable in Rn and it is not meager there. n n n Let V be the family of all Vitali sets of R , V1 be the family of all ﬁnite n n n n n unions of Vitali sets in R , V2 = {(C \ A1) ∪ A2 : C ∈ V1 ,A1 ∈ A ,A2 ∈ A } and n n n n V3 = {(C \ A1) ∪ A2 : C ∈ V1 ,A1 ∈ M0 ,A2 ∈ M0 }. 38Paper I On the families of sets without the Baire property generated by the Vitali sets

n Lemma 3.8. (i) For each element U ∈ V1 there are elements U(j), j ≤ n, of V1 Qn Qn n Qn such that U ⊂ j=1 U(j). Moreover j=1 U(j) ∈ V1 and dim( j=1 U(j)) = dim U = 0.

(ii) The family Vn of all Vitali sets of Rn as V, the family On of all open sets of n n n R as O and the family A (resp. M0 ) as M satisfy the conditions (a),(b),(c) and (d).

n Sm n Proof. (i) Let U ∈ V1 . Then U = i=1 S(i), where S(i) ∈ V for each i ≤ m. Qn Thereby S(i) = j=1 S(i, j), where S(i, j) ∈ V for each i ≤ m and j ≤ n. For Sm each j ≤ n put U(j) = i=1 S(i, j) and note that U(j) ∈ V1. Let us notice Qn Qn Sm S Qn also that U ⊂ j=1 U(j) = j=1( i=1 S(i, j)) = { j=1 S(ij, j):(i1, ··· , in) ∈ n n {1, ··· , m} } ∈ V1 . Since dim U(j) = 0 for each j ≤ n (see Proposition 3.3 (ii)), Qn we get dim( j=1 U(j)) = 0 and so dim U = 0. n n (ii) We will check only the condition (b). Let U ∈ V1 and ∅= 6 O ∈ O . It follows from (i) that there are elements U(j), j ≤ n, of V1 such that U ⊂ Qn n j=1 U(j). By the deﬁnition of the product topology on R there are non-empty Qn open sets O(j), j ≤ n, of R such that j=1 O(j) ⊂ O. In addition, by Lemma 3.2 for each j ≤ n there is a Vitali set S∗(j) of R such that S∗(j) ⊂ O(j) \ U(j). Note Qn Qn Qn Qn ∗ now that O \ U ⊃ j=1 O(j) \ j=1 U(j) ⊃ j=1(O(j) \ U(j)) ⊃ j=1 S (j) and Qn ∗ n S = j=1 S (j) ∈ V .

n Proposition 3.9. Let U ∈ V1 and E,F be meager sets (resp. sets of Lebesgue measure zero) of Rn. Then

(i) the set U ∪ E contains no diﬀerence O \ U, where O is a non-empty open set of Rn and M is a meager set (resp. a set of Lebesgue measure zero) of Rn;

(ii) the set (U \ F ) ∪ E cannot be written as (O \ M) ∪ N, where O is an open set of Rn and M,N are meager sets (resp. sets of Lebesgue measure zero) of Rn. Moreover, dim((U \ F ) ∪ E) ≤ n − 1.

Proof. The statements (i) and (ii) follow from Proposition 3.1 and Lemma 3.8 (ii). We need to give only one comment to (ii). Since (U \ F ) ∪ E ⊂ U ∪ E, it follows from (i) that the set (U \ F ) ∪ E contains no non-empty open set. Recall that each subset A of Rn with dim A = n contains a non-empty open subset of Rn [En]. Hence, dim((U \ F ) ∪ E) ≤ n − 1.

n n n n Lemma 3.10. Let V1,V2 ∈ V2 (resp. V3 ). Then V1 ∪ V2 ∈ V2 (resp. V3 ).

Proof. Straightforward.

n n Lemma 3.11. For each Vitali set S of R and x ∈ R the set Sx is also a Vitali set of Rn. 3 Main results 39

Qn Proof. Let S = j=1 S(j), where S(j), j ≤ n are Vitali sets of R, and x = Qn (x1, ··· , xn), where xj, j ≤ n, are real numbers. Note that Sx = j=1(S(j))xj .

By Lemma 3.5 the set (S(j))xj is a Vitali set of R for each j ≤ n. Hence, Sx is a Vitali set of Rn.

The following statement is evident

n n n n c Theorem 3.12. (i) V ⊂ V1 ⊂ V2 ⊂ (Bp ) .

n n (ii) For each U ∈ V1 , dim U = 0, and for each W ∈ V2 , dim W ≤ n − 1. n (Let us note that for each j ≤ n − 1 there exists a meager set Lj of R such that dim Lj = j).

n n n n (iii) The families V , V1 , V2 are invariant under translations of R .

n n (iv) The families V1 , V2 are abelian semigroups with respect to the operation of unions of sets.

0 Theorem 3.13 ([Ch]). If S is a Vitali set of Rn and (Qn) is a non-empty proper n S n 0 n c subset of Q then the set {Sr : r ∈ (Q ) } ∈ (Bp ) . Example 3.14. Let m, n be positive integers such that n ≥ 2 and m ≤ n. Let 0 also S, Q be the sets from Example 3.7. Put S(i) = S for each i ≤ n. Let us consider in Rn the set ( Qm Qn S 0 i=1 S(i) × j=m+1( {(S(j))r : r ∈ Q }), if 1 ≤ m < n; Em,n = Qn S 0 i=1( {(S(i))r : r ∈ Q }), if m = n.

Note that dim Em,n = n − m if 1 ≤ m < n, dim Em,n = n if m = n and

( S Qm Qn 0 n−m { i=1(S(i))0 × j=m+1(S(j))rj :(rm+1, ··· , rn) ∈ (Q ) }, if 1 ≤ m < n; E = 0 m,n S Qn n { j=1(S(j))rj :(r1, ··· , rn) ∈ (Q ) }, if m = n.

Put 0 n−m n 0 {(r1, ··· , rn): rk = 0, k ≤ m, (rm+1, ··· , rn) ∈ (Q ) }, if 1 ≤ m < n; (Q ) = 0 n {(r1, ··· , rn):(r1, ··· , rn) ∈ (Q ) }, if m = n.

S n 0 n 0 Note that Em,n = {Sr : r ∈ (Q ) }, and (Q ) is a non-empty proper subset n of Q . It follows now from Theorem 3.13 that the set Em,n does not possess the Baire property in Rn.

Deﬁnition 3.15. Let τ1 be a topology on a set X. A topology τ2 on X is said to be an admissible extension of τ1 if

(i) τ1 ⊂ τ2, and

(ii) τ1 is a π-base of τ2, i.e., for each non-empty element O of τ2 there is a non-empty element V of τ1 which is a subset of O. 40Paper I On the families of sets without the Baire property generated by the Vitali sets

We motivate the deﬁnition above by the following two statements.

Proposition 3.16. Let (X, τ1) be a topological space and a topology τ2 on a set X be an admissible extension of τ1. Let also S be a family of non-empty subsets of X. Then

(i) each meager set M of the space (X, τ2) is also a meager set of the space (X, τ1);

(ii) the family S as V2, τ2 as O2 and the family of all meager sets of the space (X, τ2) as M2 satisfy the conditions (a),(b),(c) and (d) whenever the families S as V1, τ1 as O1 and the family of all meager sets of the space (X, τ1) as M1 do that.

S∞ Proof. (i) Let M = i=1 Mi ⊂ X such that Intτ2 (Clτ2 Mi) = ∅ for each i. Assume that W = Intτ1 (Clτ1 Mi) 6= ∅ for some i. Let us show ﬁrst that W ⊂ Clτ2 Mi. Really, consider a point p ∈ W and an element O of τ2 containing p. Since τ1 ⊂ τ2 and W ∈ τ1, we can assume that O ⊂ W . By Deﬁnition 3.15 (ii) there is a non-empty element V of τ1 such that V ⊂ O. Note that V ∩ Mi 6= ∅ and hence

O ∩ Mi 6= ∅. So W ⊂ Clτ2 Mi. This implies that ∅= 6 W ⊂ Intτ2 (Clτ2 Mi) (recall that τ1 ⊂ τ2 and W ∈ τ1). We have a contradiction. So W = ∅ and M is a meager set of the space (X, τ1). (ii) Let us show that the condition (b) holds. Really, suppose that U is the ﬁnite union of element of S and ∅ 6= O ∈ τ2. By Deﬁnition 3.15 (ii) there is a non-empty element V of τ1 such that V ⊂ O. Since the families V1 = S, O1 = τ1 satisfy the condition (b), there is an element S of S such that S ⊂ V \U. It is easy to see that S ⊂ O \ U and the condition (b) is valid. The condition (d) follows from the statement (i) and the fact that each element of S is not meager in the space (X, τ1).

n n Let τ0 be the topology of the Euclidean space R , n ≥ 1. The following statement is a corollary of Propositions 3.1, 3.16 and Lemma 3.8 (ii).

n n n Proposition 3.17. Let U ∈ V1 and E,F be meager sets of the space (R , τ ), n n where τ is an admissible extension of τ0 . Then

(i) the set U ∪ E contains no diﬀerence O \ M, where O is a non-empty element of τ n and M is a meager set of the space (Rn, τ n); (ii) the set (U \ F ) ∪ E cannot be written as the union (O \ M) ∪ N, where O is an element of τ n and M,N are meager sets of the space (Rn, τ n).

1 Example 3.18. (a) Let τ0 = τ0 , the topology on the real line R. The following topologies τS, τc and τm on the set R are examples of admissible extension of τ0. For each point a of R a base at the point a for the topology 3 Main results 41

(i) τS consists of all sets [a, b);

(ii) τc consists of all diﬀerences (b, c)\A, where b < a < c, and A is any converging to a in the sense of τ0 sequence of points;

(iii) τm consists of all sets (b, c), where b < a < c, if a < 0, and all sets [a, b), if a ≥ 0.

Let us note that τS is the Sorgenfrey topology on R (see [En]) which is hered- itary Lindelöf and hereditarily separable. Since the space (R, τm) is evidently homeomorphic to the topological sum (R, τ0) ⊕ (R, τS), the topology τm is also hereditarily Lindelöf and hereditarily separable. The topology τc is a non-regular Hausdorﬀ topology. 2 2 (b) Let τ0 be the topology on the real x, y-plane R . The following topologies 2 2 2 2 2 τm1, τm2 and τc on the set R are examples of admissible extension of τ0 .

2 (i) τm1 coincides with the Niemytski topology for y ≥ 0 (see [2]) and the standard 2 topology τ0 for y < 0;

2 (ii) τm2 coincides with the product topology for τS × τS for x + y ≥ 0 and the 2 standard topology τ0 for x + y < 0;

2 (iii) τc is deﬁned similarly to τc.

2 2 2 Note that the topologies τm1 and τm2 are Tychonoﬀ, and the topology τc is non- regular Hausdorﬀ.

The following statement is evident.

Proposition 3.19. Let τi, νi be topologies on the set Xi such that νi is an admissible extension of τi, where i = 1, 2. Then τ1 × τ2 and ν1 × ν2 are topologies on the set X1 × X2 and ν1 × ν2 is an admissible extension of τ1 × τ2.

Example 3.20. With the help of Example 3.18 and Proposition 3.19 we can n n produce diﬀerent topologies on R which are admissible extension of τ0 . In par- k m ticular, for each integer k ≥ 1 and each integer m ≥ 0 the topology τS × τ0 on k+m k Qk R , where τS = j=1(τS)j and (τS)j = τS for each j ≤ k, is an admissible k+m extension of τ0 .

m+n m n m,n Fix m ≥ 1, n ≥ 0 and consider the space (R , τS × τ0 ). Let A (resp. m,n Bp ) be the family of all meager sets (resp. sets with the Baire property) of the m+n m n m,n m+n m,n space (R , τS × τ0 ). Let V be the family of all Vitali sets of R , V1 m,n m,n be the family of all ﬁnite unions of elements of V and V2 = {(C \ A1) ∪ A2 : m,n m,n m,n C ∈ V1 ,A1 ∈ A ,A2 ∈ A }. Theorem 3.21 ([Fo]). Let Y be a metrizable space with dim Y = 0 and ∅= 6 X ⊂ m m (R , τS ) × Y where m is a positive integer. Then dim X = 0. 42Paper I On the families of sets without the Baire property generated by the Vitali sets

m+n m n Theorem 3.22. Let m ≥ 1 and n ≥ 0. Then for the space (R , τS × τ0 ) we have the following.

m+n m,n m,n c m,n c m,n R (i) V2 ⊂ (Bp ) , where (Bp ) is the complement of Bp in 2 .

m,n (ii) For each U ∈ V1 , dim U = ind U = 0.

m,n (iii) The family V2 is an abelian semigroup with respect to the operation of union of sets.

Proof. The statement (i) follows Proposition 3.17 (ii). Let us prove the statement m,n (ii). Assume that U ∈ V1 . By Lemma 3.8 (i) there are elements U(j), j ≤ Qm+n Qm+n m,n m + n of V1 such that U ⊂ j=1 U(j) and j=1 U(j) ∈ V1 . Note that Qm+n m m Qn Qn n n j=1 U(j) ⊂ (R , τS ) × j=1 U(m + j), j=1 U(m + j) ⊂ (R , τ0 ) and dim Qn ( j=1 U(m + j)) = 0. Hence by Theorem 3.21 we get dim U = 0. Observe that the equality ind U = 0 follows from the product Theorem for ind and the property of monotonicity of ind. The proof of the statement (iii) is straightforward.

Theorem 3.23 ([Ch]). If S is a Vitali set of R and A is an inﬁnite proper subset S of Q then the set {Sr : r ∈ A} cannot be written as the union (O \ M) ∪ N, where O is an open subset of (R, τS) and M,N are meager sets of (R, τS).

Remark 3.24. It follows from Theorem 3.23 that for the set U from Example 1,0 c 3.7 we have U ∈ (Bp ) . But dim U = 0.

4 Concluding remarks

Recall (see Lemma 3.4) that the family V3 is an abelian semigroup with respect to the operation of union of sets. Moreover, V3 is invariant under translations of R.

c Question 4.1. Is the inclusion V1 ⊂ M valid?

c The positive answer on Question 4.1 would imply the inclusion V3 ⊂ M .

c c Question 4.2. Is there in the family Bp ∩ M a suﬃciently rich, invariant under translations of R subfamily F on which we can deﬁne some algebraic structure?

The positive answer on Question 4.1 would suggest the family V1 as an answer on Question 4.2.

Remark 4.3. Professor Kharazishvili kindly informed the authors that the answer to Question 4.1 is positive. We are also kindly thankful to him for his other valuable comments. References 43

References

[Ch] V.A Chatyrko, On countable unions of nonmeager sets in hereditarily Lin- delöf spaces, p-Adic Numbers, Ultrametric Analysis, and Applications, 2011, Vol.3, No.1, 1 - 6.

[En] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

[Kh] A.B. Kharazishvili, Nonmeasurable sets and functions, Vol. 195, Elsevier, 2004.

[Fo] A.A Fora, On the covering dimension of subspaces of product of Sorgenfrey lines, Proc. AMS, 1978, Vol.72, No.3, 601 - 606.

[Ox] J.C. Oxtoby, Measure and Category: Graduate texts in mathematics, Springer-Verlag, New York, Heldenberg, Berlin, 1971.

[Vi] G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, Italy, 1905.

Paper II On countable families of sets without the Baire property

Authors: Mats Aigner, Vitalij A. Chatyrko and Venuste Nyagahakwa

This paper has been published as: Mats Aigner, Vitalij A. Chatyrko and Venuste Nyagahakwa, On countable families of sets without the Baire property, Colloquium Mathematicum, 2011, Vol. 133, No.2, 179 - 187.

On countable families of sets without the Baire property

Mats Aigner∗, Vitalij A. Chatyrko∗ and Venuste Nyagahakwa∗†

Abstract

We suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (Rn, τ), where n is an integer ≥ 1 and τ is any admissible extension of the Euclidean topology of Rn (in particular, X can be a ﬁnite-dimensional separable metrizable manifold), into a countable family F of sets (dense in X and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of F does not have the Baire property in X.

Keywords: Vitali set, Baire property, admissible extension of a topology.

1 Introduction

Recall that a set A of a topological space X is said to have the Baire property in X if A = (O \ M) ∪ N, where O is an open set of X and M,N are meager sets X X of X. Let 2 be the family of all subsets of X, and Bp(X) the subfamily of 2 consisting of sets with the Baire property. It is well known that the family Bp(X) c X is a σ-algebra of sets. However, in the case when Bp(X) = 2 \ Bp(X) 6= ∅, the c c union of two sets from Bp(X) does not need to belong to Bp(X). In [Ch] (see also [CN] for generalizations) it was shown that the union of ﬁnitely many Vitali sets of the real line R (see [Vi]) contains no set of the type O \ M, where O is a non-empty open set and M is a meager one. Since each Vitali set is not meager, this easily implies that such unions do not have the Baire property.

∗Department of Mathematics, Linköping University, SE–581 83 Linköping, Sweden, E-mails: [email protected], [email protected], [email protected]. †Department of Mathematics, University of Rwanda, P.O. Box 3900 Kigali, Rwanda, E-mail: [email protected].

47 48 Paper II On countable families of sets without the Baire property

Let us note that these facts can not be extended to all countable unions of Vitali sets. It is easy to see that a set A in R is the union of a (countable) family of Vitali sets iff |A ∩ (x + Q)|= 6 0 for each x ∈ R, where Q is the set of rational numbers, i.e., A contains a Vitali set. Moreover, such a family can be chosen infinite and disjoint iff |A ∩ (x + Q)| = ℵ0 for each x ∈ R. This implies that every element of Bp(R) with non-empty interior (in particular, any non-empty open set) is the union of an infinite countable disjoint family of Vitali sets. In [Ch] the following result was proved: If S is a Vitali set of R and A is any non- S c empty proper subset of Q then U(S, A) = {r + S : r ∈ A} ∈ Bp(R). However, the set U(S, A) can contain the difference O \ M, where O is a non-empty open S 00 subset of R and M is meager. This happens iff Cl ( {(r + S) : r ∈ Q \ A}) 6= R. 00 R (we recall the operation (.) in the next section). 00 Note that for a Vitali set S such that S is dense in R and S = S (see examples S 00 in [Ch]) we have ClR( {(r + S) : r ∈ Q \ A}) = R for each non-empty proper subset A of Q. On the other hand for any Vitali set S ⊂ (−1, 1) and A = Q∩(−2, 2) we have U(S, A) ⊃ (−1, 1). In this paper, we suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (Rn, τ), where n is an integer ≥ 1 and τ is any admissible extension (see Section 3 for the definition) of the Euclidean topology of Rn (in particular, X can be a finite-dimensional separable metrizable manifold) into a countable family F of sets (dense in X and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of F does not have the Baire property in X. For the notions we refer to [E1] and [Ku].

2 Auxiliary results

We will use some notations from [Ch]. For each non-meager set R of a topological 0 space X,OR = IntX (ClX R),R = {x ∈ R∩OR : there is an open neigbourhood V 00 0 of x such that V ∩ R is meager} and R = (R ∩ OR) \ R . 0 Let us observe that by ([Ku], Theorem 1, p. 87) the set R is meager in X. 00 This implies, in particular, that the set R is non-meager in X.

Remark 2.1. Recall ([Ch], Theorem 2.1): Let X be a hereditarily Lindelöf topological space, A be a non-empty set with |A| ≤ ℵ0 and R(α) a non-meager subset S of X for each α ∈ A. Then U = {R(α): α ∈ A} ∈ Bp(X) iﬀ OR00 (α) \ U is meager in X for each α ∈ A.

Now we note that by the two sentences before this remark the equivalence above holds in any topological space X. Moreover, one can see from the proof that the necessity part is valid for any set A. 2 Auxiliary results 49

Proposition 2.2. Let X be a topological space, A a set with |A| ≥ 2 and for each

α ∈ A,Xα a non-meager subset of X. Assume also that

(i) Xα1 ∩ Xα2 = ∅ iﬀ α1 6= α2, and S (ii) O 00 is connected. α∈A Xα

0 S Then for any non-empty proper subset A of A the set Y = α∈A0 Xα does not have the Baire property.

Proof. We follow the idea of proof of ([Ch], Theorem 4.1). Assume that Y has the 0 Baire property. Then by Remark 2.1 the set O 00 \ Y is meager for each α ∈ A . Xα Further we will need the following statement. 0 0 Claim. For any α1 ∈ A and α2 ∈ A \ A we have OX00 ∩ OX00 = ∅. α1 α2

Proof. Let V = OX00 ∩ OX00 6= ∅. It follows from ([Ch], Corollary 2.1 (iii)) that α1 α2 00 00 V ∩X X ⊆ X X ∩Y = ∅ the set α2 is non-meager. Since α2 α2 and α2 by the condition 00 V ∩ X ⊂ O 00 \ Y O 00 \ Y (i), we have α2 X , where X is supposed to be meager. We α1 α1 have a contradiction which proves the Claim.

S S Put U1 = α∈A0 OX00 and U2 = α∈A\A0 OX00 . Note that the sets U1,U2 are α α S non-empty, open and by the Claim, they are disjoint. So the set O 00 = α∈A Xα U1 ∪ U2 is disconnected. We have a contradiction with (ii).

Remark 2.3. We notice that the condition (ii) from Proposition 2.2 cannot be erased. On the other hand the condition is not necessary.

(i) Let X be the subspace {0, 1} of the real line and A = {1, 2}. Set X1 = {0} and X2 = {1}. Note that O 00 = X1 and O 00 = X2. Hence O 00 ∩O 00 = ∅ X1 X2 X1 X2 and the sets X1,X2 are open in X.

(ii) Let X be the real line R, A = {1, 2} and S a Vitali set of R such that S ⊂ (0, 1). Set X1 = S and X2 = 2 + S. Note that O 00 ∩ O 00 = ∅ and the X1 X2 sets X1,X2 do not have the Baire property.

Let H(X) be the group of homeomorphisms of the space X. Then the following statement is trivial.

Lemma 2.4. Let h ∈ H(X) and A ⊂ X. Then

(i) A is meager iﬀ h(A) is meager;

0 0 00 (ii) if A is non-meager then h(OA) = Oh(A), h(A ) = (h(A)) and h(A ) = 00 (h(A)) . 50 Paper II On countable families of sets without the Baire property

Proposition 2.5. Let X be a topological space, H∗ a non-empty subset of H(X) with |H∗| ≥ 2 and A a non-meager subset of X. Assume also that

∗ (i) for any elements h1 6= h2 of H , h1(A) ∩ h2(A) = ∅, and S (ii) h∈H∗ h(OA00 ) is connected.

0 ∗ S Then for any non-empty proper subset H of H the set h∈H0 h(A) does not have the Baire property.

∗ Proof. Since for each h ∈ H the set h(A) is non-meager and h(OA00 ) = Oh(A)00 by Lemma 2.4, the statement follows from Proposition 2.2.

Proposition 2.6. Let X be a topological space of the second category, H∗ a non- empty countable subset of H(X) with |H∗| ≥ 2 and A a subset of X. Assume also that

∗ (i) for any elements h1 6= h2 of H , h1(A) ∩ h2(A) = ∅, S (ii) X \ h∈H∗ h(A) is meager, and S (iii) h∈H∗ h(OA00 ) is connected.

0 ∗ S Then for any non-empty proper subset H of H the set h∈H0 h(A) does not have the Baire property.

S Proof. Since X \ h∈H∗ h(A) is meager and the space X is of second category, the set A is nonmeager. Applying Proposition 2.5 we get the statement.

Let Q be a countable dense subgroup of the additive group of real numbers. One can consider the Vitali construction (see [Vi]) with the group Q instead of the group Q of rational numbers (cf. [Kh]). The analogue of a Vitali set with respect to Q will be called a Vitali Q-selector of R.

Example 2.7. ([Ch], Theorem 4.1 for Q = Q). Let X = R, H∗ be the group of translations of R by numbers from Q and A a Vitali Q-selector. Note that

∗ (i) for any elements h1 6= h2 of H , h1(A) ∩ h2(A) = ∅; S (ii) R \ h∈H∗ h(A) = ∅; S (iii) h∈H∗ h(OA00 ) = R is connected.

Example 2.8. ([Ch], Remark 4.2 for Q = Q)). Let X = Rn, H∗ be the group of translations of Rn by vectors with all coordinates from Q and A a Vitali Q-selector n Qn of R , that is, A = i=1 Ai, where Ai is a Vitali Q-selector of R for each i ≤ n. Note that 3 A method of constructing countable families of sets without the Baire property 51

∗ (i) for any elements h1 6= h2 of H , h1(A) ∩ h2(A) = ∅;

n S (ii) R \ h∈H∗ h(A) = ∅;

S n (iii) h∈H∗ h(OA00 ) = R is connected.

3 A method of constructing countable families of sets without the Baire property

Let τ1 be a topology on a set X. Recall ([CN], Deﬁnition 3.1) that a topology τ2 on X is said to be an admissible extension of τ1 if

(i) τ1 ⊂ τ2, and

(ii) τ1 is a π-base for τ2, i.e., for each non-empty element O of τ2 there is a non-empty element V of τ1 which is a subset of O.

Let us denote the closure (resp. the interior or the boundary) of a subset A of X in the space (X, τi) by Clτi A (resp. Intτi A or Bdτi A), where i = 1, 2.

Lemma 3.1. Let X be a set, τ1 and τ2 topologies on X such that τ2 is an admissible extension of τ1, and O is a non-empty element of τ2. Then the set

A = O \ Intτ1 O is a nowhere dense in the space (X, τ1). In particular, A is a meager set in (X, τ1).

Proof. Put V = Intτ1 O and note that V 6= ∅.

Claim. Clτ1 V ⊃ O.

Proof. Assume that W = O \ Clτ1 V 6= ∅. Since τ2 is an admissible extension of τ1 and W ∈ τ2, there is ∅= 6 U ∈ τ1 such that U ⊂ W ⊂ O. It is easy to see that U must be a subset of V . We have a contradiction which proves the Claim.

It follows from the Claim that Bdτ1 V ⊃ A. Hence, the set A is nowhere dense in the space (X, τ1).

Lemma 3.2. Let X be a set, τ1 and τ2 topologies on X such that τ2 is an admissible extension of τ1, and A ⊂ X. Assume also that A has the Baire property in the space (X, τ2). Then A has the Baire property in (X, τ1).

Proof. Suppose that A = (O \ M) ∪ N, where O is open in (X, τ2) and M,N are meager in (X, τ2). Note that by ([CN], Proposition 3.4) the sets M,N are also meager in (X, τ1). Moreover, by Lemma 3.1, the set O \ Intτ1 O is meager in

(X, τ1). Observe that A = (Intτ1 O \ M) ∪ (((O \ Intτ1 O) \ M) ∪ N). Hence, A has the Baire property in the space (X, τ1). 52 Paper II On countable families of sets without the Baire property

Let n be a positive integer. Denote by τS (resp. τ0) the Sorgenfrey topology n n (resp. the Euclidean topology) on the set R of real numbers and by τS (resp. τ0 ) Qn Qn the product i=1(τS)i (resp. i=1(τ0)i), where (τS)i = τS (resp. (τ0)i = τ0) for each i ≤ n. k m Let now k, m be non-negative integers. Note that the topology τS × τ0 of k+m k+m k+m R is an admissible extension of the Euclidean topology τ0 on R . Let k+m k m us observe that if k ≥ 1 then (R , τS × τ0 ) is disconnected and if k ≥ 2 then k+m k m (R , τS × τ0 ) is not normal. Applying Example 2.8 and Lemma 3.2 we get the following statement. Proposition 3.3. Let Q be a countable dense subgroup of the additive group of real numbers. If S is a Vitali Q-selector of Rn for some integer n ≥ 1 and A is n S c n any non-empty proper subset of Q then {r + S : r ∈ A} ∈ Bp((R , τ)), where n τ is any admissible extension of τ0 .

Remark 3.4. For the case n = 1, τ = τS and Q = Q the statement was proved in ([Ch], Corollary 4.1). Lemma 3.5. Let Y be a non-empty open subset of a space X. Then

(i) if M is a nowhere dense subset of X then M ∩ Y is a nowhere dense subset of Y .

(ii) if M is a meager subset of X then M ∩ Y is a meager subset of Y .

Proof. (i) Let V = IntY ClY (M ∩ Y ) 6= ∅. Note that V ⊂ IntY ClX (M ∩ Y ) ⊂ IntX (ClX (M ∩ Y )) ⊂ IntX ClX M. Hence, IntX ClX M 6= ∅. (ii) follows evidently from (i). Lemma 3.6. Let X be a space, Y a non-empty open subset of X and A ⊂ X. Assume also that A has the Baire property in X. Then A ∩ Y has the Baire property in Y .

Proof. Let A = (O \ M) ∪ N, where O is open in X and M,N are meager in X. Note that A ∩ Y = ((O ∩ Y ) \ (M ∩ Y )) ∪ (N ∩ Y ) and O ∩ Y is open in Y and M ∩ Y,N ∩ Y are meager in Y (by Lemma 3.5).

Remark 3.7. Let us note that the openness of the set Y in the space X in the lemmas is essential. Indeed, let X be the Euclidean plane with the x, y-axes, Y the x-axis and A a Vitali set of Y . Note that Y is a nowhere dense in X. This implies that A is also nowhere dense in X. But A does not have the Baire property in Y . Theorem 3.8. Let X be a space and Y an open subset of X which is homeomorphic to the space (Rn, τ) for some admissible extension τ of the Euclidean topology n τ0 , where n is a positive integer. Then there is an inﬁnite disjoint countable family F of sets in X such that 3 A method of constructing countable families of sets without the Baire property 53

(i) S F = X, and

0 0 (ii) for each non-empty subfamily F of F the set S F does not have the Baire property in X.

Moreover:

(a) if the set Y is dense in the space X or Z = X \ ClX Y 6= ∅ and there is a ∞ countable inﬁnite disjoint family H = {Hi}i=1 of sets dense in Z then each element of F can be chosen dense in X;

n (b) if the space X is separable metrizable, τ = τ0 and the set X \ Y is countable- dimensional then each element of F can be chosen zero-dimensional.

Proof. By Proposition 3.3 there exists an infinite disjoint countable family G = ∞ S 0 {Yi}i=1 of sets such that G = Y , and for each non-empty proper subfamily G of S 0 G the set G does not have the Baire property in Y . Then put X1 = Y1 ∪(X \Y ) ∞ and Xi = Yi, i ≥ 2. Let us notice that the countable family F = {Xi}i=1 of sets 0 is also disjoint and S F = X. Since for each non-empty proper subfamily F of 0 0 0 F there is a non-empty proper subfamily G of G such that (S F ) ∩ Y = S G , it 0 follows from Lemma 3.6 that the set S F does not have the Baire property in X. Let us now prove (a). Observe that there is a Vitali set of R which is dense in R (see [Ch], Proposition 3.3), which implies the existence of a Vitali set of Rn (see Example 2.8 for the definition) which is dense in Rn. So each set of the family G can be chosen dense in the subspace Y of X. Furthermore, if Y is dense in X then each element of the family F defined above will be dense in X.

Assume now that Z = X \ ClX Y 6= ∅ and there is a countable inﬁnite disjoint ∞ family H = {Hi}i=1 of sets dense in Z. In this case put X1 = Y1 ∪ ((X \ Y ) \ S∞ i=2 Hi) and Xi = Yi ∪ Hi, i ≥ 2. Let us notice that the countable family F = ∞ S {Xi}i=1 of sets is disjoint and dense in X. Moreover, F = X. Since for each 0 0 non-empty proper subfamily F of F there is a non-empty proper subfamily G of 0 0 0 G such that (S F ) ∩ Y = S G , it follows from Lemma 3.6 that S F does not have the Baire property in X. S∞ To prove (b), recall (cf. [E2]) that X \ Y = i=1 Zi, where the sets Z1,Z2, ··· are disjoint and dim Zi = 0 for each i. Put Xi = Yi ∪ Zi for each i ≥ 1. Since every set of the family G is also zero-dimensional it follows from the Sum Theorem for the dimension dim that the sets X1,X2, ··· are zero-dimensional.

Remark 3.9. Note that in each separable metrizable space each of whose open non-empty subset is uncountable there is an inﬁnite countable disjoint family con- ∞ sisting of dense sets. In fact, let B = {Ui}i=1 be a countable base for the space. i For each integer n ≥ 1 choose in Ui a countable inﬁnite set Ai = {aj : j ≥ 1} such S i that Ai ∩ j

Corollary 3.10. Let X be an n-dimensional separable metrizable manifold for some positive integer n. Then there exists an inﬁnite disjoint countable family F of zero-dimensional, dense in X sets such that

(i) S F = X, and

0 0 (ii) for each non-empty proper subfamily F of F the set S F does not have the Baire property in X.

n Proof. Let Y be a subset of X which is homeomorphic to R , and let Yi, i ≥ 1, be a family of dense zero-dimensional subsets of Y , which can be obtained from Example 2.8 (see also the proof of Theorem 3.8 (a)). Set B = BdX Y and V = X \ ClX Y . Assume ﬁrst that V = ∅, and note that dim B = k < n. Let us decompose the set B into k + 1 disjoint zero-dimensional subsets Bi, i ≤ k + 1. Put now Xi = Yi ∪ Bi, i ≤ k + 1, and Xi = Yi, i ≥ k + 2. Observe that the set Xi, i ≥ 1, satisfy the corollary.

Assume now that V = X \ ClX Y 6= ∅. Observe that for each point x ∈ V n there is an open neighborhood Ox of x which is homeomorphic to R . Choose in V a countable base B = {Bi : i ≥ 1} for open sets with (n − 1)-dimensional S∞ boundaries. Set V0 = V \ i=1 BdV (Bi). Notice that dim V0 = 0 and each open S∞ non-empty subset of V0 is uncountable. Decompose the set i=1 BdV (Bi) into n zero-dimensional disjoint sets V1, ··· ,Vn. Note that V0, ··· ,Vn are dense in V . i By Remark 3.9 we can decompose V0 into sets V0 , i ≥ 1, dense in V0. Observe i that V0 , i ≥ 1, are dense in V and zero-dimensional. Now the argument can be ﬁnished as in the ﬁrst case.

4 Concluding remarks

It is known that no Bernstein set of R has the Baire property. We will show this fact with the help of Proposition 2.2. Recall that a subset A of the real line R is called a Bernstein set if for each Cantor set C ⊂ R we have A ∩ C 6= ∅ and (R \ A) ∩ C 6= ∅.

Lemma 4.1. For each meager subset M of R there exists a Cantor set C such that M ∩ C = ∅. In particular, each Bernstein set B of R is non-meager and OB00 = R.

S∞ Proof. We can suppose that M = i=1 Mi, where Mi is a nowhere dense and closed in R for each i. Hence, the set N = R \ M is topologically complete. If N contains a non-degenerate interval [a, b] then N contains a Cantor set. Otherwise, N is zero-dimensional and nowhere locally compact. This implies that References 55

N is homeomorphic to the space P of irrational number (cf. [Mi]). Hence, N must contains a Cantor set. 00 00 00 Since B \ B is meager (see [[Ch], Proposition 2.1]) and B ⊂ B, the set B must be dense in R by the main statement of this lemma. Hence OB00 = R.

Proposition 4.2. No Bernstein set of R has the Baire property.

Proof. Let M be a Bernstein set of R. Note that R \ M is also a Bernstein set and the sets M, \ M are disjoint and non-meager. Moreover, O 00 = O 00 = . R M R\M R By Proposition 2.2 the set M does not have the Baire property.

Acknowledgments: This research was partly supported by Sida.

References

[Ch] V.A. Chatyrko, On countable unions of nonmeager sets in hereditarily Lin- delöf spaces, p-Adic Numbers and Ultrametric Ana. Appl., 2011, Vol.3, No.1, 1 - 6.

[CN] V.A. Chatyrko and V. Nyagahakwa, On the families of sets without the Baire property generated by the Vitali sets, p-Adic Numbers and Ultrametric Ana. Appl., 2011, Vol.3, No.2, 100 - 107.

[E1] R. Engelking, General Topology, 2nd ed., Heldermann Verlag, Berlin, 1989.

[E2] R. Engelking, Theory of Dimensions, Finite and Inﬁnite, Heldermann, Lemgo, 1995.

[Kh] A.B Kharazishvili, Nonmeasurable Sets and Functions, Elsevier, Amsterdam, 2004.

[Ku] K. Kuratowski, Topology, Volume I, PWN, Warszawa, and Academic Press, New York, 1966.

[Mi] J. van Mill, The Inﬁnite-Dimensional Topology of Functions Spaces, Elsevier, Amsterdam, 2001.

[Vi] G.Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, 1905.

Paper III The algebra of semigroups of sets

Authors: Mats Aigner, Vitalij A. Chatyrko and Venuste Nyagahakwa

This paper has been published as: Mats Aigner, Vitalij A. Chatyrko and Venuste Nyagahakwa, The algebra of semigroups of sets, Mathematica Scandinavica, 2015, Vol.116, No.2, 161-170.

The algebra of semigroups of sets

Mats Aigner∗, Vitalij A. Chatyrko∗ and Venuste Nyagahakwa∗†

Abstract

We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For n > 1 we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of Rn without the Baire property.

Keywords: Semigroup of sets, Baire property, Vitali set, nonmeasurable in the Lebesgue sense.

1 Introduction

An interesting extension of the family M of all meager subsets of the real line R, as well as the family O of all open subsets of R, in the family P(R) of all subsets of R, is the family Bp of all sets possessing the Baire property. The property is a classical notion which is related to the thesis of R. Baire. Recall that B ∈ Bp if there are an O ∈ O and an M ∈ M such that B = O∆M.

It is well known that the family Bp is a σ-algebra of sets invariant under c homeomorphisms of the real line R, and the complement Bp = P(R) \ Bp of Bp in P(R) is not empty (for example, each Vitali set S of R [Vi] is an element of c c Bp). Moreover, there are elements of Bp with a natural algebraic structure (see c [Kh1] for subgroups of the additive group R, which are elements of Bp). c In [CN] Chatyrko and Nyagahakwa looked for subfamilies of the family Bp which have some algebraic structures. They proved that the family V1 of all ﬁnite unions of Vitali sets of R and its extension V2 which elements are all sets of type A∆B, where A ∈ V1 and B ∈ M, are semigroups of sets (i.e. families of sets closed under the ﬁnite unions) invariant under translations of real line R and consisting ∗Department of Mathematics, Linköping University, SE–581 83 Linköping, Sweden, E-mails: [email protected], [email protected], [email protected]. †Department of Mathematics, University of Rwanda, P.O. Box 3900 Kigali, Rwanda, E-mail: [email protected].

59 60 Paper III The algebra of semigroups of sets

c of zero-dimensional sets and V2 ⊂ Bp. Furthermore, Chatyrko and Nyagahakwa extended the result to the Euclidean spaces Rn, where n is any positive integer. In this paper we pay attention to the algebra of semigroups of sets. We look at the behavior of semigroups of sets under several operations. Then we suggest some applications. First, we show that the results from [CN] can be obtained by the use of the theory. Moreover, we can suggest many different semigroups of sets c in Bp. After that for each n > 1 we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of Rn without the Baire property.

2 Auxiliary notions

Recall that a non-empty set S is called a semigroup if there is an operation α : S × S −→ S such that α(α(s1, s2), s3) = α(s1, α(s2, s3)). The semigroup S is called abelian if α(s1, s2) = α(s2, s1). Let X be a set and P(X) be the family of all subsets of X. In the paper we will be interested in subsets S of P(X) such that for each A, B ∈ S we have A ∪ B ∈ S. It is evident that such a family of sets is an abelian semigroup with the respect to the operation of union of sets (in brief, a semigroup of sets). S Let A ⊂ P(X). Put SA = { i≤n Ai : Ai ∈ A , n ∈ N}. Note that SA is a semigroup of sets. Recall that a non-empty collection of sets I ⊂ P(X) is called an ideal of sets if I is a semigroup of sets and if A ∈ I and B ⊂ A then B ∈ I . Put IA = {B ∈ P(X): there is A ∈ SA such that B ⊂ A}. Note that IA is an ideal of sets.

For x ∈ R denote by Tx the translation of R by x, i.e., Tx(y) = y + x for each y ∈ R. If A is a subset of R and x ∈ R, we denote Tx(A) by Ax. The equivalence relation E on R is deﬁned as follows. For x, y ∈ R, let xEy iﬀ x−y ∈ Q, where Q is the set of rational numbers. Let us denote its equivalence classes by Eα, α ∈ I. It is evident that |I| = c (continuum), and for each α ∈ I and each x ∈ Eα,Eα = Qx. Let us also note that every equivalence class Eα is dense in R. Recall [Vi] that a Vitali set of R is any subset S of R such that |S ∩ Eα| = 1 for each α ∈ I, and each Vitali set neither possess the Baire property in R nor it is measurable in the Lebesgue sense. For other notions and notations we refer to [En] and [Ku].

3 Semigroups of sets and ideals of sets

Let A , B ⊂ P(X). Put A ∪ B = {A ∪ B : A ∈ A ,B ∈ B}, A ∆B = {A∆B : A ∈ A ,B ∈ B} and A ∗ B = {(A \ B1) ∪ B2 : A ∈ A ; B1,B2 ∈ B}. However, A ∩ B denotes the intersection of A and B, i.e., the family of common elements of A and B. It is evident that A ∪ B = B ∪ A and A ∆B = B∆A . Since 3 Semigroups of sets and ideals of sets 61

A ∪ B = (A \ B) ∪ B = (B \ A) ∪ A, we have A ∪ B ⊂ A ∗ B and A ∪ B ⊂ B ∗ A . Moreover, if A , B are both semigroups of sets or both ideals of sets then the family A ∪ B is of the same type. On the other hand, as we will see in the following examples, in general for given semigroups of sets A , B, the families A ∆B, A ∗ B, B ∗ A do not need to be semigroups of sets and none of the statements A ∆B ⊆ A ∪ B, A ∆B ⊇ A ∪B, A ∆B ⊆ A ∗B, A ∆B ⊇ A ∗B, A ∗B ⊆ B ∗A needs to hold. Moreover, one of the families A ∗ B, B ∗ A can be a semigroup of sets while the other is not. Example 3.1. Let |X| ≥ 2 and A be a non-empty proper subset of X. Put B = X \ A, A = {A, X} and B = {B,X}. Note that A = SA , B = SB and the families A ∪ B = {X}, A ∆B = {∅, A, B, X}, A ∗ B = {B,X}, B ∗ A = {A, X} are semigroups of sets. Moreover, none of the following inclusions A ∆B ⊆ A ∪ B, A ∆B ⊆ A ∗ B, A ∗ B ⊆ B ∗ A and B ∗ A ⊆ A ∗ B holds.

Example 3.2. Let X = {1, 2, 3, 4},A1 = {1, 3},A2 = {2, 4},B1 = {1, 2},B2 = {3, 4},C = {1, 4},D = {2, 3}, A = {∅,A1,A2} and B = {∅,B1,B2}. Note that SA = {∅,A1,A2,X} and SB = {∅,B1,B2,X}. Moreover, we have SA ∪ SB = c c c c c {∅,A1,A2, B1,B2, {1} , {2} , {3} , {4} ,X} (here Y denotes the complement of a set Y in the set X), SA ∆SB = {∅,A1,A2,B1,B2,C,D,X} and SA ∗ SB = SB ∗SA = P(X)\{C,D}. It is easy to see that the inclusions SA ∗SB ⊆ SA ∆SB and SA ∪ SB ⊆ SA ∆SB do not hold. We note also that none of the families SA ∆SB, SA ∗ SB and SB ∗ SA is a semigroup of sets. In fact, A1,D ∈ SA ∆SB c but A1∪D = {4} ∈/ SA ∆SB, and {1}, {4} ∈ SA ∗SB but {1}∪{4} = C/∈ SA ∗SB.

Example 3.3. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9},A1 = {1, 2, 4, 5, 7, 8}, A2 = {2, 3, 5, 6, 8, 9}, B1 = {1, 2, 3, 4, 5, 6}, B2 = {4, 5, 6, 7, 8, 9}, A = {A1,A2}, B = {∅,B1,B2}. Note that SA = {A1,A2,X} and SB = {∅,B1,B2,X}. First we will show that the family SA ∗ SB is not a semigroup of sets. It is enough to prove that the set C = ((A1 \ B1) ∪ ∅) ∪ ((A2 \ B2) ∪ ∅) ∈/ SA ∗ SB. Note that C = (A1 \ B1) ∪ (A2 \ B2) = {2, 3, 7, 8}. Assume that C ∈ SA ∗ SB. Thus C = (S1 \ S2) ∪ S3 for some S1 ∈ SA and S2,S3 ∈ SB. Since |C| = 4, we have S3 = ∅. Let S1 = A1. Then |S1 \ S2| is either 2 (if S2 is B1 or B2), 0 (if S2 = X) or 6 (if S2 = ∅). We have a contradiction. If S1 = A2, we also have a contradiction by a similar argument as above. Assume now that S1 = X. Then |S1 \S2| is either 3 (if S2 is B1 or B2), 0 (if S2 = X) or 9 (if S2 = ∅). We have again a contradiction that proves the statement. c c c c Further note that SB ∗ SA = {A1,A2, {1} , {3} , {7} , {9} ,X} = SA ∪ SB. Hence, the family SB ∗ SA is a semigroup of sets. Proposition 3.4. Let S be a semigroup of sets and I be an ideal of sets. Then the family S ∗ I is a semigroup of sets.

0 00 Proof. In fact, let Si ∈ S and Ii ,Ii ∈ I , i = 1, 2. Proceed as follows: U = 0 00 0 00 0 0 00 00 00 00 ((S1 \ I1) ∪ I1 ) ∪ ((S2 \ I2) ∪ I2 ) = (S1 \ I1) ∪ (S2 \ I2) ∪ (I1 ∪ I2 ). Put I2 = I1 ∪ I2 62 Paper III The algebra of semigroups of sets

0c 0c cc 0c c 0c c c and continue: U = ((S1 ∩I1 )∪(S2 ∩I2 )) ∪I2 = ((S1 ∩I1 ) ∩(S2 ∩I2 ) ) ∪I2 = c 0 c 0 c c c c 0 c 0 0 0 c ((S1 ∪ I1) ∩ (S2 ∪ I2)) ∪ I2 = ((S1 ∩ S2) ∪ (S1 ∩ I2) ∪ (S2 ∩ I1) ∪ (I1 ∩ I2)) ∪ I2. c 0 c 0 0 0 c c c c Put I1 = (S1 ∩ I2) ∪ (S2 ∩ I1) ∪ (I1 ∩ I2) and note that U = ((S1 ∩ S2) ∩ I1) ∪ I2 = c ((S1 ∪ S2) ∩ I1) ∪ I2 = ((S1 ∪ S2) \ I1) ∪ I2. It is easy to see that S1 ∪ S2 ∈ S and I1,I2 ∈ I . Hence, U ∈ S ∗ I .

Let (X, τ) be a topological space and M(X, τ) be the family of meager subsets of (X, τ). It is easy to see that the family τ is a semigroup of sets and M(X, τ) is an ideal of sets (in fact, a σ-ideal of sets). The family Bp(X, τ) of sets with the Baire property is deﬁned as the family τ∆M(X, τ). It is well known that τ∆M(X, τ) = τ ∗ M(X, τ). In fact, this equality is a particular case of the following general statement.

Proposition 3.5. Let S be a semigroup of sets and I be an ideal of sets. Then

(a) S ∗ I = S∆I ⊃ S ∪ I = I ∗ S ⊃ S;

(b) (S ∗ I ) ∗ I = S ∗ I , I ∗ (I ∗ S) = I ∗ S.

Proof. (a) Note that for any set S ∈ S and for any set I ∈ I we have S∆I = (S \ I) ∪ (I \ S) ∈ S ∗ I ,S ∪ I = S∆(I \ S) ∈ S∆I ,S ∪ I = (I \ S) ∪ S ∈ I ∗ S and S = S ∪ ∅ ∈ S ∪ I . Thus, S ∗ I ⊃ S∆I ⊃ S ∪ I ⊃ S and I ∗ S ⊃ S ∪ I . Observe also that for any sets S1,S2 ∈ S and any sets I1,I2 ∈ I we have (S1 \ I1) ∪ I2 = S1∆I ∈ S∆I , where I = ((I1 ∩ S1) \ I2) ∪ (I2 \ S1), and (I1 \ S1) ∪ S2 ∈ S ∪ I . There by, S ∗ I ⊂ S∆I and I ∗ S ⊂ S ∪ I .

(b) Let S ∈ S and I1,I2,I3,I4 ∈ I . Observe that (((S \ I1) ∪ I2) \ I3) ∪ I4 = (S \ (I1 ∪ I3)) ∪ ((I2 \ I3) ∪ I4) ∈ S ∗ I . Hence (S ∗ I ) ∗ I ⊂ S ∗ I . The opposite inclusion is evident.

Let I1,I2,I3 ∈ I and S1,S2,S3,S4 ∈ S. Note that (I1 \ ((I2 \ S1) ∪ S2)) ∪ ((I3 \ S3) ∪ S4) = ((I1 \ ((I2 \ S1) ∪ S2)) ∪ (I3 \ S3)) ∪ S4 = I ∪ S4 ∈ I ∗ S, where I = (I1 \ ((I2 \ S1)) ∪ S2)) ∪ (I3 \ S3). Hence I ∗ (I ∗ S) ⊂ I ∗ S. The opposite inclusion is evident.

Corollary 3.6. Let S be a semigroup of sets and I be an ideal of sets. Then

(a) the families S∆I , I ∗ S are semigroups of sets;

(b) (I ∗ S) ∗ I = I ∗ (S ∗ I ) = S ∗ I .

Proof. We will only show (b). Note that (1) S ∗ I = (S ∗ I ) ∗ I ⊃ (I ∗ S) ∗ I ⊃ S ∗ I ; (2) S ∗ I = (S ∗ I ) ∗ I ⊃ I ∗ (S ∗ I ) ⊃ S ∗ I .

The following statement is evident. 3 Semigroups of sets and ideals of sets 63

Corollary 3.7. Let I1, I2 be ideals of sets. Then the family I1 ∗ I2 is an ideal of sets. Moreover, I1 ∗ I2 = I2 ∗ I1 = I1∆I2 = I1 ∪ I2.

Example 3.8. Let X = {1, 2},A = X,B = {1},C = {2}, A = {A}, B = {B}. Note that SA = {A}, SB = {B}, IB = {∅,B}, SA ∗ IB = {A, C} and IB ∗ SA = {A}. Thus, in general, none of the following statements is valid: S ∗ I = I ∗ S, S ∗ I ⊃ I , the family S ∗ I is an ideal of sets or I ∗ S is an ideal of sets, even if S is a semigroup of sets and I is an ideal of sets.

The next statement is useful in the search of pairs of semigroups without common elements.

Proposition 3.9 (see [CN], Proposition 3.1). Let I be an ideal of sets and A , B ⊂ P(X) such that:

(a) A ∩ I = ∅ (i.e. A and I have no common elements);

(b) for each element U ∈ SA and each non-empty element B ∈ B there is an element A ∈ A such that A ⊂ B \ U.

Then

(1) for each element I ∈ I , each element U ∈ SA and each non-empty element B ∈ B we have (U ∪ I)c ∩ B 6= ∅;

(2) for each elements I1,I2 ∈ I , each element U ∈ SA and each non-empty c element B ∈ B we have (U ∪ I1) ∩ (B \ I2) 6= ∅;

(3) for each elements I1,I2,I3,I4 ∈ I , each element U ∈ SA and each element V ∈ SB we have (U \ I1) ∪ I2 6= (V \ I3) ∪ I4, i.e., (SA ∗ I ) ∩ (SB ∗ I ) = ∅.

Proof. Our proof is very close to the proof of [[CN], Proposition 3.1]. (1) Assume that U ∪ I ⊃ B for some non-empty element B ∈ B. By (b) there is A ∈ A such that A ⊂ B \U. Note that A ⊂ (U ∪I)\U ⊂ I. But this contradicts (a).

(2) Assume that U ∪ I1 ⊃ B \ I2 for some non-empty element B ∈ B and some element I2 ∈ I . Note that U ∪ (I1 ∪ I2) = (U ∪ I1) ∪ I2 ⊃ (B \ I2) ∪ I2 ⊃ B. But this contradicts (1).

(3) Assume that (U \I1)∪I2 = (V \I3)∪I4 for some elements U ∈ SA ,V ∈ SB and I3,I4 ∈ I . If V = ∅, then (U \ I1) ∪ I2 = I4 and so U ⊂ I1 ∪ I4. But this contradicts (a). Hence V 6= ∅. Note that there is a non-empty element B ∈ B such that B ⊂ V . Further observe that U ∪ I2 ⊃ (U \ I1) ∪ I2 = (V \ I3) ∪ I4 ⊃ B \ I3. But this contradicts (2).

Example 3.10 ([CN]). (a) The family V of all Vitali sets of R as A , the family O of all open sets of R as B and the family M of all meager sets of R 64 Paper III The algebra of semigroups of sets

as I satisfy the conditions of Proposition 3.9. Note that SV = V1, SO = O, V1 ∗ M = V2 and O ∗ M = Bp (the notations are from the Introduction). Hence, V2 ∩ Bp = ∅.

(b) Consider the Euclidean space Rn for some n > 1. A Vitali set of Rn is any Qn set S = j=1 S(j), where S(j) is a Vitali set of R for each j = 1, 2, ..., n. The family Vn of all Vitali sets of Rn as A , the family On of all open sets of Rn as B and the family Mn of all meager sets of Rn as I , satisfy the conditions n of Proposition 3.9. Let V1 be the family of all ﬁnite unions of Vitali sets of n n n n n n R , V2 = V1 ∗ M and Bp be the family of all sets of R with the Baire n n n n n n n property. Note that SVn = V1 , SOn = O , Bp = O ∗ M and V2 ∩ Bp = ∅. n m There is even a generalization of the result for the products R × RS , where RS is the Sorgenfrey line (see [En] for the deﬁnition).

4 Applications

In ([CN], Theorem 3.2) one can ﬁnd the following statements about the families n n n V , V1 and V2 , where n ≥ 1.

n n n n c (i) V ⊂ V1 ⊂ V2 ⊂ (Bp ) .

n n (ii) For each U ∈ V1 , dim U = 0, and for each W ∈ V2 , dim W ≤ n − 1.

n n n n (iii) The families V , V1 , V2 are invariant under translations of R . n n (iv) The families V1 , V2 are semigroups of sets.

4.1 Two nested families of semigroups of sets

n n It follows easily from Corollary 3.6 and Proposition 3.5 that the family M ∗ V1 n n is another semigroup of sets invariant under translations of R such that V1 ⊂ n n n M ∗ V1 ⊂ V2 . The following statement extends the variety of semigroups of sets n without the Baire property based on the family V1 . n,k n−1 n,k n−1 Theorem 4.1. Let n > 1. Then there are two ﬁnite families {L }k=0 , {R }k=0 of pairwise distinct semigroups of sets invariant under translations of the Euclidean space Rn such that

(a) for each 0 ≤ k ≤ n − 2 we have L n,k ⊂ L n,k+1 and Rn,k ⊂ Rn,k+1,

(b) for each L ∈ L n,k and R ∈ Rn,k we have dim L ≤ k and dim R ≤ k and n,k n,k there are L0 ∈ L and R0 ∈ R such that dim L0 = dim R0 = k where 0 ≤ k ≤ n − 1,

(c) for each 0 ≤ k ≤ n − 1 we have L n,k ⊂ Rn,k but Rn,k−1 does not contain L n,k, 4 Applications 65

n,n−1 n n,n−1 n n,n−1 n n n,n−1 n n (d) R ⊂ V2 but R 6= V2 , L ⊂ M ∗ V1 but L 6= M ∗ V1 n n n,0 n n,0 n n,0 and M ∗ V1 does not contain R , V1 ⊂ L but V1 6= L .

Proof. For each 0 ≤ k < n let us consider the family Fk of all closed k-dimensional n subsets of R . Note that every family Fk is a semigroup of sets and the inclusion

IFk ⊂ IFk+1 holds for each 0 ≤ k ≤ n − 2. Since every element of IFn−1 is n n nowhere dense in the Euclidean space R we have IFn−1 ⊂ M . n,k n n,k n For each 0 ≤ k < n put R = V1 ∗ IFk and L = IFk ∗ V1 . The point (a) is evident. It follows from Proposition 3.4 and Corollary 3.6 that the families Rn,k, L n,k are semigroups of sets for each 0 ≤ k < n. It is also clear that the families Rn,k, L n,k consist of sets which are invariant under translations of Rn and which have dimension dim ≤ k. Since for each Vitali set S of Rn the union S∪Ik = (Ik \ S) ∪ S = (S \ Ik) ∪ Ik, where Ik is any subset of Rn homeomorphic to the k- dimensional cube [0, 1]k, belongs to both families L n,k, Rn,k and dim(S ∪Ik) = k, we have (b). Note that Proposition 3.5 implies the inclusion of (c), and (b) implies that L n,k−1 6= L n,k, Rn,k−1 6= Rn,k and that the family Rn,k−1 cannot contain the family L n,k. On the other hand for each Vitali set S of Rn the diﬀerence S \{p}, n n where p ∈ S, cannot belong to the family M ∗ V1 but it belongs to the family Rn,0. Hence, L n,k 6= Rn,l for each 0 ≤ k, l < n − 1. Note that Rn,n−1 ⊂ n n,n−1 n n n n,0 V2 , L ⊂ M ∗ V1 and V1 ⊂ L . In order to ﬁnish the proof of (d) let us n recall (see [[CN], Lemma 3.4]) that for each element U ∈ V1 there are elements Qn n V1, ··· ,Vn ∈ V1 such that U ⊂ i=1 Vi. This easily implies that no element of V1 can contain a countable subset of Rn consisting of points with rational coordinates. Thus the set Cn ∪ S = (Cn \ S) ∪ S ∈ L n,0, where C is the standard Cantor set n n of [0, 1] and S is any Vitali set of R , is not any element of V1 , and the set n n n n Q ∪ S = (Q \ S) ∪ S ∈ M ∗ V1 , where Q is the set of all rational numbers of R and S is any Vitali set of Rn, is not an element of Rn,n−1. This completes the proof of (d).

4.2 Supersemigroups based on the Vitali sets

Let Q be a countable dense subgroup of the additive group of the real numbers. One can consider the Vitali construction [Vi] with the group Q instead of the group Q of rational numbers (cf. [Kh1]). The analogue of a Vitali set with respect to the group Q will be called a Vitali Q-selector of R. One can introduce in the same way as above a Vitali Q-selector of Rn, n ≥ 1 and the corresponding families n n n n n n,k n,k V (Q), V1 (Q), M ∗ V1 (Q), V2 (Q), L (Q), R (Q), where 0 ≤ k < n. Note that similar statements as in Subsection 4.1 are valid for the families. Let F be the family of all countable dense subgroups of the additive group sup sup sup of real numbers. Set V = {V : V ∈ V(Q),Q ∈ F }, V1 = SVsup and V2 = sup V1 ∗ M. 66 Paper III The algebra of semigroups of sets

It is easy to see that

sup (i) for each Q ∈ F we have V2(Q) ⊂ V2 .

sup sup (ii) V1 and V2 are semigroups of sets invariant under translations of R.

One can even show that for each Q ∈ F we have V(Q) ⊂ Vsup but V(Q) 6= Vsup, sup sup sup V1(Q) ⊂ V1 but V1(Q) 6= V1 . We do not know if V2(Q) 6= V2 for each Q ∈ F . sup We will call the family V1 the supersemigroup of sets based on the Vitali sets.

sup Lemma 4.2. For any set U ∈ V1 and any non-empty open set O of R, there is a set V ∈ Vsup such that V ⊂ O \ U.

Sn Proof. Let U = i=1 Vi, where Vi ∈ V(Qi) and Qi ∈ F , i = 1, ··· , n. Note that the statement is valid when Q1 = ··· = Qn (see [[CN] Lemma 3.1]). Now we will Pn Pn consider the general case. Put Q = i=1 Qi = { i=1 qi : qi ∈ Qi} and note that Q ∈ F .

Claim 4.3. For each x ∈ R we have |Qx ∩ (O \ U)| ≥ 1. (In fact |Qx ∩ (O \ U)| = ℵ0).

Proof. For n = 1 the statement evidently holds ([[CN] Lemma 3.1]). Let n ≥ 2. Let Oi, i ≤ n, be non-empty open sets of R such that x + O1 + ··· + On = {x + x1 + ··· + xn : xi ∈ Oi, i ≤ n} ⊂ O. For each i ≤ n choose n + 1 diﬀerent points qi(j), j ≤ n + 1, of Oi ∩ Qi. ∗ j j Let now Qi = {qi : j ≥ 1}, i = 1 ≤ n, and qi = qi(j), i ≤ n; j ≤ n + 1. Observe that for each i ≤ n and each j1, ··· , jbi, ··· , jn (the notation ba means that a is not j1 k jn there) the set {x + q1 + ··· + qi + ··· + qn : k ≥ 1} consists of countably many diﬀerent points (a coset of Qi) and only one of them belongs to Vi.

Consider now an n-dimensional digital box B = {(j1, ··· , jn): ji ≤ n + 1, i ≤ n}. Note that |B| = (n + 1)n and call the elements of B by cells. Put in each cell j1 jn (j1, ··· , jn) of B the sum x + q1 + ··· + qn . Fix i ≤ n and observe that each interval I(j1, ··· jbi, ··· , jn) = {(j1, ··· , k, ··· , jn): k ≤ n+1} of cells contains at most one element of Vi. So the whole box B contains n−1 n−1 at most (n + 1) elements of Vi. Summarizing we have at most n(n + 1) elements of U in the box B. Since (n + 1)n > n(n + 1)n−1 for n ≥ 2, there are points p in B which are not elements of U. But such p must be element of the set Qx ∩ O by our choice. The claim is proved.

Let us ﬁnish the proof of the Lemma. For each equivalence class Qx choose a point from the set Qx ∩ (O \ U). The set of such points is a Vitali Q-selector V of R such that V ⊂ O \ U. References 67

sup c Theorem 4.4. (a) V2 ⊂ Bp.

sup (b) for each A ∈ V2 we have dim A = 0.

sup (c) for each Q ∈ F we have V2(Q) ⊂ V2 .

sup (d) V2 is a semigroup of sets and it is invariant under translations of R.

Proof. (a) and (b) follow Lemma 4.2 and Proposition 3.9. (c) and (d) were observed earlier in (i) and (ii) of this section.

Remark 4.5. (a) Considering diﬀerent ideals of sets in the real line R (the ideal of ﬁnite sets, the ideal of countable sets, the ideal of closed and discrete sets, the ideal of nowhere dense sets, etc) we can produce many semigroups of sets c in Bp by the use of the operation ∗ and the semigroups V1(Q),Q ∈ F and sup V1 .

(b) Let us note that one can deﬁne supersemigroups of sets based on the Vitali sets in Rn, n ≥ 2, by a similar argument as above.

4.3 A nonmeasurable case

In [Kh2] Kharazishvili proved that each element U of the family V1 is nonmeasurable in the Lebesgue sense. Let N be the family of all measurable sets in the Lebesgue sense on the real line R and let N0 ⊂ N be the family of all sets of the Lebesgue measure zero. Recall that the family N0 is an ideal of sets (in fact, a σ-ideal). It follows from Propositions 3.4 and 3.5 that the families V1, N0 ∗ V1 and V1 ∗N0 are three diﬀerent semigroups of sets invariant under translations of R and V1 ⊂ N0 ∗ V1 ⊂ V1 ∗ N0. We have the following generalization of Kharazishvili’s result.

Proposition 4.6. Each element of the family V1 ∗ N0 is nonmeasurable in the Lebesgue sense.

Proof. In fact, let A ∈ V1 ∗ N0 and assume that A ∈ N . By Proposition 3.5 there are an U ∈ V1 and an N ∈ N0 such that A = U∆N. It is known that if A1,A2 are sets such that A1 ∈ N and the set A1∆A2 is of the Lebesgue measure zero then the set A2 must belong to the family N (see [CK]). But A∆U = (U∆N)∆U = N, hence U ∈ N . This is a contradiction with [Kh2]. So A/∈ N .

sup Question 4.7. Is each element U of the family V1 nonmeasurable in the Lebesgue sense?

Acknowledgements. The authors would like to thank the referee for his (her) valuable comments. 68 Paper III The algebra of semigroups of sets

References

[CK] M. Capiński, E. Kopp, Measure, Integral and Probability, Second edition, Springer, Undergraduate Mathematics Series, Springer-Verlag, London, 2004.

[En] R. Engelking, General topology, Heldermann Verlag, Berlin, 1989.

[Kh1] A. B Kharazishvili, Nonmeasurable sets and functions, Elsevier, Amster- dam, 2004.

[Kh2] A. B Kharazishvili, Measurability properties of Vitali sets, American Math- ematical Monthly, 2011, Vol.118, No.8, 693 - 703.

[Ku] K. Kuratowski, Topology, Volume I, PWN, Warszawa and Academic Press, New York, 1966.

[Vi] G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, 1905. Paper IV Vitali selectors in topological groups and related semigroups of sets

Authors: Vitalij A. Chatyrko and Venuste Nyagahakwa

This paper has been published as: Vitalij A. Chatyrko and Venuste Nyagahakwa, Vitali selectors in topological groups and related semigroups of sets, Questions and Answers in General Topology, 2015, Vol.33, No.2, 93 - 102.

Vitali selectors in topological groups and related semigroups of sets

Vitalij A. Chatyrko∗ and Venuste Nyagahakwa∗†

Abstract

In an abelian topological group G having a countable dense subgroup Q, we study the family V(Q) of Vitali selectors of G based on Q. Thus the family SV(Q) of ﬁnite unions of Vitali selectors is invariant under translations of G and its elements do not possess the Baire property in G. Moreover, the family {G \ U : U ∈ SV(Q)} is a base for a T1 (but not T2)-topology τ(Q) on G. We consider properties of the topological space (Q, τ(Q)). In particular, we show that for any countable dense subgroup Q1 and Q2 of G, the topological spaces (G, τ(Q1)) and (G, τ(Q2)) are homeomorphic.

Keywords: Vitali set, Baire property, abelian topological group, semigroup of sets, hyperconnected space.

1 Introduction

Let R be the set of real numbers, P(R) be the family of all subsets of R, (R, τE) be the real line (i.e. the set R endowed with the topology τE deﬁned by all open intervals), and M be the family of all meager subsets of (R, τE). An interesting extension of M as well as τE in P(R) is the family Bp of sets possessing the Baire property.

Recall (cf. [Ox]) that B ∈ Bp iff there are O ∈ τE and M ∈ M such that B = O∆M. It is well known that the family Bp is a σ-algebra of sets. In particular, Bp is closed under finite unions and finite intersections of sets. It is easy to see that Bp is also invariant under the action of the group H ((R, τE)) of homeomorphisms of (R, τE) onto itself. ∗Department of Mathematics, Linköping University, SE–581 83 Linköping, Sweden, E-mails: [email protected], [email protected]. †Department of Mathematics, University of Rwanda, P.O. Box 3900 Kigali, Rwanda, E-mail: [email protected].

71 72 Paper IV Vitali selectors in topological groups and related semigroups of sets

c Let us recall that the complement Bp = P(R) \ Bp of Bp in P(R) is not c empty (for example, each Vitali set S of R [Vi] is an element of Bp). It is easy to c see that the family Bp is also invariant under the action of the group H ((R, τE)) c but unlike to Bp the family Bp is neither closed under finite unions nor finite intersections of sets. c One can ask: Are there in Bp families of sets which are invariant under the action of an infinite subgroup of H ((R, τE)) and on which we can define some algebraic structure? The following simple observations could give an answer to the question.

Let G be a non-trivial subgroup of H ((R, τE)). Assume that A is a subfamily c of Bp such that A is invariant under the action of G. If for each n ≥ 2 and each Sn c A1,A2, ··· ,An ∈ A we have i=1 Ai ∈ Bp, then the family SA consisting of all finite unions of elements of A is an abelian semigroup of sets w.r.t the binary operation ” ∪ ” of union of sets. It is easy to see that SA is invariant under the c action of G and SA ⊆ Bp. In [Ch], Chatyrko showed that each union of finitely many Vitali sets is an c element of Bp. It is easy to see that the family V (resp. SV ) of all Vitali sets (resp. finite unions of Vitali sets) is invariant under the action of the group τ(R) of translations of R. Hence, SV is an abelian semigroup of sets w.r.t the operation c "∪", which is invariant under the action of τ(R) and SV ⊆ Bp.

In [CN] Chatyrko and Nyagahakwa proved that the family SV ∆M = {U∆M : U ∈ SV ,M ∈ M} is also a semigroup w.r.t the operation "∪", it is invariant under c the action of τ(R) and SV ∆M ⊆ Bp. Furthermore, in [ACN], Aigner, Chatyrko and Nyagahakwa generalized the method from [CN] and by the use of elementary group theory and elements of dimension theory, found many distinct semigroups c n of sets in Bp and its analogs based on the Euclidean spaces R , n ≥ 2. What can be done next? In this paper we replace the Euclidean spaces Rn, n ≥ 1, by abelian Hausdorff nonmeager uncountable topological groups G having countable dense subgroups Q. Then for each Q we consider the family V(Q) of all Vitali Q-selectors of G (which are closely related to the Vitali sets on the real line and which are called Q- selectors in [Kh]) and the semigroup SV(Q) consisting of finite unions of elements of V(Q). Furthermore, we show that SV(Q) defines a new topology τ(Q) on G (which is T1 but not T2). We study properties of the spaces (G, τ(Q)). In particular, we prove that for any countable dense subgroups Q1 and Q2 of G the topological spaces (G, τ(Q1)) and (G, τ(Q2)) are homeomorphic. 2 Auxiliary notions 73

2 Auxiliary notions

2.1 Baire property

Let X be a topological space and A ⊆ X. By ClX A (resp. IntX A) we will denote the closure (resp. the interior) of A in X.

The set A is called nowhere dense (resp. meager) in X if IntX (ClX A) = ∅ (resp. A is the union of countably many nowhere dense sets in X). The set A is called nonmeager in X or of second category if it is not meager in X. The set A is said to possess the Baire property in X iﬀ there are an open set O and a meager set M such that A = O∆M, where ∆ is the operation of symmetric diﬀerence. Recall that the family Bp(X) of sets possessing the Baire property in X, is a σ-algebra of sets.

2.2 Semigroups and ideals of sets

Let X be a non-empty set and P(X) the family of subsets of X.

A collection S ⊆ P(X) is called a semigroup of sets if for every S1,S2 ∈ S we have S1 ∪ S2 ∈ S. A collection I ⊆ P(X) is called an ideal of sets if I is a semigroup of sets and if A ∈ I and B ⊆ A then B ∈ I . S Example 2.1. Let A ⊆ P(X). The family SA = { i≤n Ai ∈ P(X): Ai ∈ A , n ∈ N} is a semigroup of sets. The family IA = {B ∈ P(X): there is A ∈ SA such that B ⊆ A} is an ideal of sets.

A way to get a new semigroup of sets from an old one and an ideal of sets is described in the following statement.

Proposition 2.2 ([ACN]). Let S be a semigroup of sets and I be an ideal of sets. Then

(a) the families S∆I and S ∪ I are semigroups of sets;

(b) S∆I ⊃ S ∪ I ⊃ S;

(c) (S∆I )∆I = S∆I , (S ∪ I ) ∪ I ) = S ∪ I , where A ∪ B = {A ∪ B : A ∈ A ,B ∈ B} and A ∆B = {A∆B : A ∈ A ,B ∈ B} for any A , B ⊆ P(X).

Example 2.3. Let (X, τ) be a topological space with the topology τ, and M(X, τ) be the family of meager subsets of (X, τ). It is easy to see that the collection τ is a semigroup of sets, the collection M(X, τ) is an ideal of sets (in fact, a σ-ideal of sets) and Bp(X, τ) = τ∆M(X, τ). 74 Paper IV Vitali selectors in topological groups and related semigroups of sets

The next statement is useful in the search of pairs of semigroups of sets without common elements.

Proposition 2.4 ([ACN]). Let I be an ideal of sets and A , B ⊆ P(X) such that

(a) A ∩ I = ∅;

(b) for each element U ∈ SA and each non-empty element B ∈ B there is an element A ∈ A such that A ⊆ B \ U.

Then (SA ∆I ) ∩ (SB∆I ) = ∅ (by C ∩ D we denote the intersection of C , D ⊆ P(X), i.e., the family of common elements of C and D).

2.3 A topology deﬁned by a semigroup

Let X be a non-empty set and B ⊆ P(X) such that

(a) for each x ∈ X there exists an element B ∈ B containing x;

(b) for any B1,B2 ∈ B and x ∈ B1 ∩ B2 there exists B3 ∈ B such that x ∈ B3 ⊆ B1 ∩ B2.

0 0 It is well known that the family τB = {∪B : B ⊆ B} is a topology on X for which the family B is a base. Let S ⊆ P(X) be a semigroup of sets. We say that S satisfies the property (∗) if for each x ∈ X there exists an S ∈ S such that x∈ / S. The next evident statement describes how to get a topology from a semigroup of sets. Proposition 2.5. Let S ⊆ P(X) be a semigroup of sets which satisfies the property (∗). Then the family BS = {X \ S : S ∈ S} satisfies the conditions (a) and

(b), and hence the family BS is a base for the topology τBS on X.

For other notions, notations and facts we refer to [En] and [Ox].

3 Vitali selectors in an abelian topological group G and related semigroup of sets

All considered topological groups are abelian, Hausdorﬀ, of the second category, without isolated points and having a countable dense subgroup. Note that each such a topological group is uncountable.

Example 3.1. Simple examples of our topological groups are the real numbers R with the addition x+y : x, y ∈ R, as the group operation endowed with the natural 3 Vitali selectors in an abelian topological group G and related semigroup of sets 75 topology of the real line; the set T of complex numbers of length 1 with the product z1 · z2 : z1, z2 ∈ T, as the group operation endowed with the subspace topology from the complex plane, ﬁnite folders Rn, Tm and products Rn × Tm, n, m ≥ 1, where the group operations and topology are deﬁned naturally.

Let G be a topological group and Q be a countable dense subgroup of G. For simplicity, below we will write the group operation on G additively.

3.1 A semigroup of Vitali selectors of G

For x ∈ G by Tx we will denote the translation of G by x, i.e., Tx(y) = y + x for each y ∈ G. If A is a subset of G and x ∈ G, we will denote Tx(A) by Ax.

Remark 3.2. Let us note that each translation Tx, where x ∈ G, is a homeomor- phism of the topological group G into itself, and the family τ(G) of translations of G is a subgroup of the group H (G) of homeomorphisms of G onto itself.

The equivalence relation E(Q) of G is deﬁned as follows. For x, y ∈ G, let xE(Q)y iﬀ x − y ∈ Q. Let us denote the equivalence classes of this relation by Eα(Q), α ∈ I. Note that for each α ∈ I and each x ∈ Eα(Q), Eα(Q) = Qx. Hence, every equivalence class Eα(Q) is dense in G and |Eα(Q)| = ℵ0. It is also evident that |I| = |G|. A Vitali Q-selector of G (or shortly, a Vitali selector of G if the group G can not be confused or its structure is nonessential) is any subset V of G such that |V ∩ Eα(Q)| = 1 for each α ∈ I. Note that |V | = |G|.

Let V(Q) be the family of all Vitali Q-selectors of G. The family SV(Q) is deﬁned as in Example 2.1. The following statements can be proved by a standard way.

Proposition 3.3. The family V(Q) (resp. SV(Q)) is invariant under translations of G.

Proposition 3.4. Let V ∈ V(Q). Then we have the following.

S (a) G = q∈Q Vq, and all Vq, q ∈ Q, are disjoint and homeomorphic to each other;

(b) Each V is nonmeager in G.

Lemma 3.5. For each element U ∈ SV(Q) and each non-empty open subset O of G there is an element V ∈ V(Q) such that V ⊆ O \ U.

Sn Proof. Let U = i=1 Vi, where Vi ∈ V(Q) for each i. Since |Eα(Q) ∩ O| = ℵ0 and |Eα(Q) ∩ U| ≤ n, we have Eα(Q) ∩ (O \ U) 6= ∅ for each equivalence class Eα(Q). So we can choose a Vitali Q-selector V such that V ⊆ O and V ∩ U = ∅. 76 Paper IV Vitali selectors in topological groups and related semigroups of sets

Let τ be the topology on the group G. Note that τ is a semigroup of sets of G and τ = Sτ . By the use of Lemma 3.5 and Proposition 2.4 we get the following. Proposition 3.6. Let I be an ideal of sets in G such that V(Q) ∩ I = ∅. Then (SV(Q)∆I ) ∩ (τ∆I ) = ∅.

Let Im be the ideal of meager subsets of G. Recall that τ∆Im is the family of subsets of G with the Baire property in G.

Corollary 3.7. The family SV(Q)∆Im consists of subsets of G which do not possess the Baire property in G. In particular, each element of SV(Q) does not have the Baire property in G.

Proof. By Proposition 3.4 we have V(Q) ∩ Im = ∅. The rest follows from Propo- sition 3.6.

Question 3.8. Let Ui ∈ SV(Q) and fi ∈ H (G), where i = 1, 2, ··· , n ≥ 2. Is the Sn union i=1 fi(Ui) without the Baire property in G?

3.2 Diversity of semigroups of Vitali selectors

Let Q1 and Q2 be subgroups of the group G. Deﬁne Q1 + Q2 = {q1 + q2 : qi ∈ Qi, i = 1, 2} ⊆ G. Note that Q1 +Q2 is a subgroup of G and Q1, Q2 are subgroups of Q1 + Q2. Moreover,

(i) if each Qi, i = 1, 2 is countable then the subgroup Q1 + Q2 is also countable;

(ii) if one of the groups Qi, i = 1, 2 is dense in the topological group G then so is Q1 + Q2.

Denote by F the family of all countable and dense subgroups of G.

Lemma 3.9. For each Q1 ∈ F there exists Q2 ∈ F such that Q1 ⊆ Q2 and Q1 6= Q2.

Proof. Let a ∈ G \ Q1 (recall that |G| > ℵ0). Note that A = {na : n is an integer} (the set A can be ﬁnite) is a subgroup of G. Put Q2 = Q1 + A and note that Q2 is a countable dense subgroup of G. Moreover, Q1 ⊆ Q2 and Q1 6= Q2.

Proposition 3.10. Let Q1,Q2 ∈ F such that Q1 ⊆ Q2 and Q1 6= Q2. Then

SV(Q1) ∩ V(Q2) = ∅. In particular, V(Q1) ∩ V(Q2) = ∅.

Proof. Assume that SV(Q1) ∩ V(Q2) 6= ∅ and let V ∈ SV(Q1) ∩ V(Q2). Consider L an equivalence class Eα(Q2). Note that Eα(Q2) is the disjoint union {Eβ(Q1): β ∈ Bα} of some equivalence classes Eβ(Q1), β ∈ Bα, and |Bα| > 1 (because of

Q1 6= Q2). Since V ∈ V(Q2) (resp. V ∈ SV(Q1)) we have |V ∩ Eα(Q2)| = 1 (resp. L |V ∩ Eβ(Q1)| ≥ 1 for each β ∈ Bα and hence |V ∩ Eα(Q2)| = |V ∩ {Eβ(Q1): β ∈ Bα}| ≥ |Bα| > 1). This is a contradiction. 3 Vitali selectors in an abelian topological group G and related semigroup of sets 77

Let Q1,Q2 ∈ F such that Q1 ⊆ Q2, Q1 6= Q2 and Q2/Q1 be the factor group of Q2 by Q1. Then we have either 1 < |Q2/Q1| < ∞ or |Q2/Q1| = ℵ0.

Proposition 3.11. For each Q1 ∈ F there exists Q2 ∈ F such that Q1 ⊆ Q2, Q1 6= Q2 and |Q2/Q1| = ℵ0.

0 Proof. Set Q1 = Q1. It follows from Lemma 3.9 by induction that for each positive i i−1 i i−1 i integer i there exists Q1 ∈ F such that Q1 ⊆ Q1 and Q1 6= Q1. Put Q2 = S∞ i i=1 Q1 and note that Q2 ∈ F . Moreover, Q1 ⊆ Q2, Q1 6= Q2 and |Q2/Q1| = ℵ0.

Proposition 3.12. Let Q1,Q2 ∈ F such that Q1 ⊆ Q2 and Q1 6= Q2. Then the following statements are valid:

(i) If |Q2/Q1| < ∞ then SV(Q1) ⊆ SV(Q2) and SV(Q1) 6= SV(Q2).

(ii) If |Q2/Q1| = ℵ0 then SV(Q1) ∩ SV(Q2) = ∅.

Proof. (i) First, let us show that V(Q1) ⊆ SV(Q2). This will imply the inclusion

SV(Q1) ⊆ SV(Q2). In fact, consider V ∈ V(Q1) and let |Q2/Q1| = n for some integer n > 1. Note that each equivalence class Eα(Q2), α ∈ I, is the disjoint union L {Eβ(Q1): β ∈ Bα} of some equivalence classes Eβ(Q1), β ∈ Bα, and |Bα| = n. Since for each α we have |V ∩ Eα(Q2)| = n, enumerate the set V ∩ Eα(Q2) as α α α α {x1 , ··· , xn}. Set V1 = {x1 : α ∈ I}, ··· ,Vn = {xn : α ∈ I}. Note that Sn V1, ··· ,V2 ∈ V(Q2) and V = i=1 Vi ∈ SV(Q2). Since SV(Q1) ∩ V(Q2) = ∅ (by

Proposition 3.10), we get SV(Q1) ⊆ SV(Q2) and SV(Q1) 6= SV(Q2).

(ii) Assume that SV(Q1) ∩ SV(Q2) 6= ∅ and V ∈ SV(Q1) ∩ SV(Q2). Consider an L equivalence class Eα(Q2). Recall that Eα(Q2) is the disjoint union {Eβ(Q1): β ∈ Bα} of some equivalence classes Eβ(Q1), β ∈ Bα, and |Bα| = ℵ0. Since

V ∈ SV(Q2) (resp. SV(Q1)) we have |V ∩ Eα(Q2)| < ∞ (resp. |V ∩ Eβ(Q1)| ≥ 1 for L each β ∈ Bα, and hence |V ∩ Eα(Q2)| = |V ∩ {Eβ(Q1): β ∈ Bα}| = ℵ0). This is a contradiction.

Corollary 3.13. For each Q1 ∈ F there exists Q2 ∈ F such that Q1 ⊆ Q2,

Q1 6= Q2 and SV(Q1) ∩ SV(Q2) = ∅. In particular, the family {SV(Q) : Q ∈ F } of all semigroups of Vitali Q-selectors has no element which contains all others.

Proof. It follows from Proposition 3.11 that for each Q1 ∈ F there exists Q2 ∈ F such that Q1 ⊆ Q2, Q1 6= Q2 and |Q2/Q1| = ℵ0. Proposition 3.12 (ii) implies the rest.

Remark 3.14. Similarly to [ACN] one can deﬁne a family Vsup = {V : V ∈ V(Q),Q ∈ F } and semigroups SVsup , SVsup ∪ Im, SVsup ∆Im such that

(i) SVsup ⊆ SVsup ∪ Im ⊆ SVsup ∆Im; 78 Paper IV Vitali selectors in topological groups and related semigroups of sets

(ii) SV(Q) ⊆ SVsup for each Q ∈ F ;

(iii) the family SVsup ∆Im consists of sets without the Baire property in G.

3.3 Topologies on G deﬁned by semigroups of Vitali selectors

Let Q ∈ F , B(Q) = BSV(Q) , τ(Q) = τB(Q) and G(Q) be the topological space (G, τ(Q)). We will list some properties of G(Q).

Proposition 3.15. Let A ⊆ G. Then A is closed in G(Q) iﬀ sup{|A ∩ Eα(Q)| : α ∈ I} < ℵ0 or A = G.

Proof. ” =⇒ ” Assume that A is closed in G(Q) and A 6= G. Consider x ∈ G \ A. Sn Sn Note that there exists V1, ··· ,Vn ∈ V(Q) such that x∈ / i=1 Vi and A ⊆ i=1 Vi. Sn Moreover, sup{|A ∩ Eα(Q)| : α ∈ I} ≤ sup{|( i=1 Vi) ∩ Eα(Q)| : α ∈ I} ≤ n < ℵ0.

” ⇐= ” Assume that sup{|A ∩ Eα(Q)| : α ∈ I} < ℵ0 and put n = sup{|A ∩ Eα(Q)| : α ∈ I}. Note that G \ A 6= ∅. Consider any x ∈ G \ A. For each α α α ∈ I, let {x1 , ··· , xn} denotes the set A ∩ Eα(Q) (possibly with overlapping) if α α A ∩ Eα(Q) 6= ∅ and let x1 = ··· = xn ∈ Eα(Q) \{x}, otherwise. Then the sets α Sn Vi = {xi : α ∈ I}, i = 1, ··· , n are Vitali Q-selectors such that x∈ / i=1 Vi and Sn A ⊆ i=1 Vi. Thus A is closed in G(Q).

The following two statements are evident.

Corollary 3.16. G(Q) is a T1-space.

Corollary 3.17. For each x ∈ G, any countable inﬁnite subset of Qx is dense in the space G(Q).

Recall [SS] that a topological space X (or the topology on X) is called hyperconnected if for every two open non-empty subsets U, V of X we have U ∩ V 6= ∅. It is easy to see that each hyperconnected topological space is connected and non- Hausdorﬀ. Moreover, each open non-empty subset of such a space is dense in the space.

Corollary 3.18. The space G(Q) cannot be represented as the union of two closed proper subsets. In particular, G(Q) is hyperconnected.

Proof. Assume that G = A1 ∪ A2, where each Ai is closed in G(Q) and Ai 6= G. Let ni = sup{|Ai ∩ Eα(Q)| : α ∈ I} < ℵ0, i = 1, 2. It is easy to see that sup{|(A1 ∪ A1) ∩ Eα(Q)| : α ∈ I} ≤ n1 + n2 < ℵ0. Since |Eα(Q)| = ℵ0 for each α ∈ I, we have a contradiction.

Remark 3.19. Recall that each T1-topological group must be Hausdorﬀ. Hence G(Q) is not a topological group. 3 Vitali selectors in an abelian topological group G and related semigroup of sets 79

Qb Recall [Kn] that the box product α∈A Xα of any inﬁnite system of nondegenerated regular T1-spaces Xα, α ∈ A, is disconnected.

Proposition 3.20. Let Xα, α ∈ A, be an inﬁnite system of hyperconnected spaces. Qb Then the box product α∈A Xα is hyperconnected (in particular, is connected).

Qb Proof. In fact, consider two non-empty open subsets O and V of α∈A Xα. Choose 0 Qb 0 Qb two open basic subsets O = α∈A Oα and V = α∈A Vα, where Oα and Vα are 0 0 non-empty open subsets of Xα such that O ⊆ O and V ⊆ V . Since Xα, α ∈ A, are hyperconnected, there is pα ∈ Oα ∩ Vα. Note that the point {pα : α ∈ A} 0 0 belongs to both sets O and V .

The following statement is evident. Qb Corollary 3.21. The box product α∈A G(Qα), where Qα ∈ F for each α ∈ A, is connected.

Question 3.22. Is there a family Xα, α ∈ A, of Hausdorﬀ non-degenerated con- Qb nected spaces such that the box product α∈A Xα is connected? Proposition 3.23. Each non-empty closed subsets A of G(Q) distinct from G is discrete in G(Q).

Proof. Indeed, let A be a proper non-empty closed subset of G(Q) and x ∈ A. Note that the set A \{x} is also closed in G(Q) by the use of Proposition 3.15. Hence the subspace A of G(Q) has the discrete topology.

Corollary 3.24. (a) No element of SV(Q) with the subspace topology from G(Q) is Lindelöf, in particular, the space G(Q) is not Lindelöf.

(b) Each open non-empty subset O of the topological group G (in particular, G) is a disjoint union of countably many closed discrete subsets of the space G(Q) of cardinality |G|.

Proof. (a) Recall only that for each U ∈ SV(Q) we have |U| = |G| > ℵ0.

(b) Note that for each α ∈ I we have |Eα(Q) ∩ O| = ℵ0, and let Eα(Q) ∩ O = 1 2 i {xα, xα, ···}. Set Vi = {xα : α ∈ I} for each i = 1, 2, ··· . Note that Vi, i = 1, 2, ··· S∞ are disjoint, Vitali Q-selectors and O = i=1 Vi (recall that each Vitali Q-selectors V has |V | = |G| > ℵ0).

Let ind and Ind be the small and large inductive dimensions deﬁned for general spaces (see [Pe]).

Corollary 3.25. ind G(Q) = Ind G(Q) = 1 80 Paper IV Vitali selectors in topological groups and related semigroups of sets

Proof. Note that for any open non-empty set O distinct from G we have Bd O = G\O 6= ∅. Since the set G\O is discrete (see Proposition 3.23), we have ind Bd O = Ind Bd O = 0. This implies the statement.

Now we will consider the following question.

Question 3.26. Let Q1,Q2 ∈ F . What is the relationship between the subfamilies τ(Q1) and τ(Q2) of P(G) (resp. the topological spaces G(Q1) and G(Q2)?

Proposition 3.27. Let Q1,Q2 ∈ F such that Q1 ⊆ Q2,Q1 6= Q2 and |Q2/Q1| = ℵ0. Then there exists a subset A of G such that A is closed in G(Q1) but A is neither closed nor open in G(Q2). In particular, τ(Q1) * τ(Q2).

L∞ Proof. Note that the set Q2 ⊆ G is the disjoint union i=1(Q1)xi of countable many translations of Q1. Put A = {xi : i = 1, 2, ···}. It easily follows from Proposition 3.15 that the set A is closed in G(Q1) but A is neither closed nor open in G(Q2).

It is easy to see that τ(Q1) ⊆ τ(Q2) whenever SV(Q1) ⊆ SV(Q2).

Question 3.28. Let Q1,Q2 ∈ F . Is the inclusion SV(Q1) ⊆ SV(Q2) valid whenever τ(Q1) ⊆ τ(Q2)?

Example 3.29. Let X be a set such that |X| > ℵ0. Consider a decomposition L X = α∈I Xα of X into disjoint countable subsets Xα, α ∈ I. Note that |I| = |X|. Deﬁne a topology τC on X as follows: A proper subset A of X is said to be closed in the topological space (X, τC ) iﬀ sup{|A ∩ Xα| : α ∈ I} < ℵ0.

Theorem 3.30. Let Q ∈ F . Then the space G(Q) is homeomorphic to the space (X, τC ) with |X| = |G|. In particular, for any Q1,Q2 ∈ F the space G(Q1) and G(Q2) are homeomorphic.

Proof. For each α ∈ I consider a bijection bα : Eα(Q) −→ Xα. Deﬁne a mapping b : G −→ X by b(x) = bα(x) whenever x ∈ Eα(Q). Note that b is a homeomor- phism between G(Q) and (X, τC ) (apply Proposition 3.15).

Acknowledgments: The authors would like to thank the referee for his (her) good advice which helped essentially to improve the presentation of the results.

References

[ACN] M. Aigner, V.A. Chatyrko and V. Nyagahakwa, The algebra of semigroups of sets, Mathematica Scandinavica, 2015, Vol.116, No.2, 161-170.

[Ch] V.A. Chatyrko, On Vitali Sets and their unions, Matematicki Vesnkik, 2011, Vol.63, No.2, 87 - 92. References 81

[CN] V.A. Chatyrko and V. Nyagahakwa, On the families of sets without the Baire property generated by the Vitali sets, P-Adic Numbers, Ultrametric Analysis, and Applications, 2011, Vol.3, No.2, 100 - 107.

[En] R. Engelking, General topology, Heldermann Verlag, Berlin, 1989.

[Kh] A.B Kharazishvili, Nonmeasurable sets and functions, Elsevier, Amsterdam, 2004.

[Kn] C.J. Knight, Box topology, Quart. J. Math. Oxford Ser., 1964, Vol.2, No. 15, 41 - 54.

[Ox] J.C. Oxtoby, Measure and category, Springer-Verlag, New York Helderberg Berlin, 1971.

[Pe] A.R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, 1975.

[SS] L.A. Steen and J.A. Seebach, Jr., Counterexamples in topology, Springer- Verlag, 1978.

[Vi] G.Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, 1905.

Paper V Sets with the Baire property in topologies formed from a given topology and ideals of sets

Authors: Vitalij A. Chatyrko and Venuste Nyagahakwa

This paper has been accepted and it is available as: Vitalij A. Chatyrko and Venuste Nyagahakwa, Sets with the Baire property in topologies formed from a given topology and ideals of sets, 2017.

Sets with the Baire property in topologies formed from a given topology and ideals of sets

Vitalij A. Chatyrko∗ and Venuste Nyagahakwa∗†

Abstract

Let X be a set, τ1, τ2 topologies on X and Bp(X, τi) the family of all subsets of X possessing the Baire property in (X, τi), i = 1, 2. In this paper we study conditions on τ1 and τ2 that imply a relationship (for example, inclusion or equality) between the families Bp(X, τ1) and Bp(X, τ2). We are mostly interested in the case where the topology τ2 is formed with the help of a local function deﬁned by the topology τ1 and an ideal of sets I on X. We also consider several applications of the local function deﬁned by the usual topology on the reals and the ideal of all meager sets there, for proving some known facts.

Keywords: Baire property, π-compatible topologies, Admissible extension, Local function, ∗-topology,Vitali set and Bernstein set.

1 Introduction

Let X be a non-empty set, P(X) the family of all subsets of X, τ a topology on X and Im the family of all meager sets of the topological space (X, τ). An interesting collection of subsets of X extending τ, as well as Im, is the family Bp(X, τ) of all subsets of X possessing the Baire property in (X, τ) (see the next part for a deﬁnition). It is well known that the family Bp(X, τ) is a σ-algebra of sets which is invariant under homeomorphisms of the space (X, τ). Let us note that Bp(X, τtriv) = {∅,X} and Bp(X, τdisc) = P(X), where τtriv (resp. τdisc) is the trivial (resp. discrete) topology on the set X. It is easy to see that for

85 Paper V Sets with the Baire property in topologies formed from a given topology and 86 ideals of sets any set X there is no topology τ distinct from τtriv such that Bp(X, τ) = {∅,X}. But there exist a set X and a topology τ on X distinct from τdisc such that Bp(X, τ) = P(X), and there are a set X and a topology τ on X distinct from τtriv for which Bp(X, τ) 6= P(X) (see Remark 2.1). One can pose the following

Problem 1.1. Let τi, i = 1, 2, be topologies on a set X.

(A) What relationship between the topologies implies a relationship (for example, inclusion or equality) between the families of nowhere dense sets (resp. meager sets or sets with the Baire property)?

(B) Does a relationship (for example, inclusion or equality) between the families of nowhere dense sets (resp. meager sets or sets with the Baire property) imply some relationship between the topologies?

It is clear that if τ1 = τ2 then Bp(X, τ1) = Bp(X, τ2). However, the inclusion τ1 ⊆ τ2 does not need to imply the inclusion Bp(X, τ1) ⊆ Bp(X, τ2) (see Example 3.23). In this paper we will mostly consider the case when the larger topology τ2 in the inclusion is obtained from the topology τ1 by a way described below. In [JH] Janković and Hamlett introduced a method to define new topologies from old ones via ideals of sets. Namely, for any space (X, τ) and any ideal I of subsets of X, they considered two operators ( · )∗ and Cl∗( · ) on the family P(X) (see the next part for definitions). The first operator ( · )∗ was a local function in the sense of Hayashi [Ha]. The second operator Cl∗( · ), naturally obtained from ( · )∗, was a Kuratowski closure operator (cf. [Ku]). Then they defined a new topology τ ∗(I ) on X by a standard way as the family {U ⊆ X : Cl∗(X \ U) = X \ U} and studied its properties. In particular, they showed that

(i) τ ⊆ τ ∗(I ), and

∗ ∗ (ii) if I = {∅} (resp. I = P(X)) then τ (I ) = τ (resp. τ (I ) = τdisc); moreover, if I1 and I2 are ideals of sets on X such that I1 ⊆ I2 then ∗ ∗ ∗ τ (I1) ⊆ τ (I2); thus for every ideal I of sets on X we have τ ⊆ τ (I ) ⊆ τdisc.

∗ ∗ It is easy to see that Bp(X, τ (I )) ⊆ Bp(X, τ (P(X)) for any ideal I of sets on X. So we will study the relationship (in the sense of inclusions or equalities) ∗ between the families Bp(X, τ) and Bp(X, τ (I )) as well as between the fami- ∗ ∗ lies Bp(X, τ (I1)) and Bp(X, τ (I2)), where I1 and I2 are ideals of sets on ∗ ∗ ∗ X. Since the equality τ (I1) = τ (I2) implies the equality Bp(X, τ (I1)) = ∗ Bp(X, τ (I2)), we shall also be interested in the cases when the ideals I1 and I2 produce the same topology on X. Furthermore, we shall consider some applications of the local function defined by the usual topology on the reals and the ideal Im of meager sets there for proving already known facts. In particular, we shall 2 Preliminary notions 87 show that the local function can be used to prove the result in [CN] stating that the family SV ∆Im = {U∆M : U ∈ SV ,M ∈ Im}, where SV is the family of all finite unions of Vitali subsets of R and ∆ is the symmetric difference operation, consists of sets without the Baire property on the real line.

2 Preliminary notions

2.1 Ideal of sets and the Baire property

Let X be a non-empty set and P(X) the family of all subsets of X. Recall that a non-empty collection I ⊆ P(X) of sets is called an ideal of sets on X if it satisﬁes the following conditions:

(i) If A ∈ I and B ∈ I then A ∪ B ∈ I .

(ii) If A ∈ I and B ⊆ A then B ∈ I .

If the ideal of sets I is closed under countable unions of sets then it is called a σ-ideal of sets on X. Let (X, τ) be a topological space and let A be a subset of X. The notation Cl(A) (resp. Int(A)) will stand for the closure (resp. the interior) of A in (X, τ). Recall that a set A is said to be nowhere dense in (X, τ) if for each non-empty element U ∈ τ there exists a non-empty element W ∈ τ such that W ⊆ U and W ∩ A = ∅ (in short, Int Cl(A) = ∅). A set A is called meager in (X, τ) if A is the union of countably many nowhere dense sets in (X, τ). A set A is called nonmeager in (X, τ) if it is not meager in (X, τ). The family In (resp. Im) of all nowhere dense (resp. meager) sets in (X, τ) forms an ideal (resp. a σ-ideal) of sets on X. Recall that a set A is said to possess the Baire property in (X, τ) if it can be written in the form A = (O \ M) ∪ N where O ∈ τ and M,N ∈ Im. The family Bp(X, τ) of sets with the Baire property is a σ-algebra of sets which contains all Borel and all meager sets in (X, τ) and which is invariant under homeomorphisms of the space (X, τ).

Remark 2.1. Note that for the real line (R, τ), i.e., for the set R of real numbers endowed with a topology τ generated by all open intervals, we have Bp(R, τ) 6= P(R) (for example, any Vitali set V of R [Vi] is an element of P(R) \ Bp(R, τ)). However, for the set Q of rational numbers endowed with the subspace topology

τQ from the real line, we have Bp(Q, τQ) = P(Q). Paper V Sets with the Baire property in topologies formed from a given topology and 88 ideals of sets 2.2 Local functions and Kuratowski operators via ideals of sets, ∗-topologies

We follow [JH]. Let (X, τ) be a topological space, I an ideal of sets on X and let A be a subset of X. For a point x ∈ X, let N (x) = {U ∈ τ : x ∈ U}. A local function of the set A with the respect to an ideal I of sets and the topology τ, is the set A∗(I , τ) = {x ∈ X : A ∩ U/∈ I for every U ∈ N (x)}. The notation A∗(I ) will be used instead of A∗(I , τ) if it is clear what topology is used. It is evident that if I ∈ I then I∗(I ) = ∅. The following theorem establishes basic properties of the local function.

Theorem 2.2 (cf. [JH]). Let (X, τ) be a topological space, I and J ideals of sets on X and let A and B be subsets of X. Then

(i) If A ⊆ B then A∗(I ) ⊆ B∗(I ); (iv) If I ∈ I then (A ∪ I)∗(I ) = (A \ I)∗(I ) = A∗(I );

(ii) (A ∪ B)∗(I ) = A∗(I ) ∪ B∗(I ); (v) If I ⊆ J then A∗(J ) ⊆ A∗(I );

(iii) A∗(I ) = Cl(A∗(I )) ⊆ Cl(A); (vi) [A∗(I )]∗(I ) ⊆ A∗(I ).

For every A ⊆ X put Cl∗(A) = A ∪ A∗(I ). It is easy to see that Cl∗( · ) is a Kuratowski closure operator. So the family {U ⊆ X : Cl∗(X \ U) = X \ U} is a topology on the set X denoted by τ ∗(I ). This topology is called the ∗-topology obtained from the original topology τ and the ideal of sets I . Recall that the family β(τ, I ) = {O \ I : O ∈ τ, I ∈ I } is a base for the topology τ ∗(I ). It is clear that τ ⊆ τ ∗(I ). Moreover, every element of the ideal I is closed and discrete in the space (X, τ ∗(I )).

2.3 Some natural relation between topologies and ideals

Recall [DGR] that an ideal I is called codense in a topological space (X, τ) if τ ∩ I = {∅}.

Theorem 2.3 (cf. [Mo]). Let (X, τ) be a topological space and I be an ideal of sets on X. The following statements are equivalent:

(i) τ ∩ I = {∅}. (iv) If U ∈ τ then U ∗(I ) = Cl(U). (ii) If I ∈ I then Int(I) = ∅. (iii) X = X∗(I ). (v) For every U ∈ τ, U ⊆ U ∗(I ). 3 Two topologies on the same set and the Baire property 89

For other notions and facts we refer to [Ku] and [Ox].

3 Two topologies on the same set and the Baire property

3.1 π-compatible topologies

Deﬁnition 3.1. Let τ, σ be topologies on a set X. We call the topologies π- compatible if τ is a π-network for σ (i.e. for each non-empty element O of σ there is a non-empty element V of τ which is a subset of O) and vice versa.

The following statement is evident.

Proposition 3.2. Let Xα, α ∈ A be disjoint sets and for each α ∈ A the topologies τα, σα be π-compatible on Xα. Then

L L S (i) the sum topologies {τα : α ∈ A} and {σα : α ∈ A} on the set {Xα : α ∈ A} are π-compatible. Q Q Q (ii) the product topologies {τα : α ∈ A} and {σα : α ∈ A} on the set {Xα : α ∈ A} are π-compatible.

Now we will explain why π-compatible topologies are interesting.

Lemma 3.3. Let τ and σ be π-compatible topologies on X. Then the spaces (X, τ) and (X, σ) have the same families of nowhere dense sets (resp. meager sets).

Proof. Assume that N is a nowhere dense set in the space (X, τ). We will show that N is also a nowhere dense set in the space (X, σ). Consider a non-empty set Oσ ∈ σ. Since τ is a π-network for σ, there exists a non-empty set Oτ ∈ τ such that Oτ ⊆ Oσ. Since N is a nowhere dense set in the space (X, τ), there is a non-empty set Vτ ∈ τ such that Vτ ⊆ Oτ and Vτ ∩ N = ∅. Since σ is a π-network for τ, there exists a non-empty set Wσ ∈ σ such that Wσ ⊆ Vτ . Note that Wσ ⊆ Oσ and Wσ ∩ N = ∅. Hence, N is a nowhere dense set in the space (X, σ). The inverse implication can be proved by a similar argument. The case of meager sets is obvious.

Theorem 3.4. Let τ and σ be π-compatible topologies on X. Then the spaces (X, τ) and (X, σ) have the same families of sets with the Baire property.

Proof. We will show that Bp(X, τ) ⊆ Bp(X, σ). Since the families of nowhere dense sets for the spaces (X, τ) and (X, σ) coincide by Lemma 3.3, it is enough to prove that each element of τ is the union of an element of σ and a nowhere dense set. We use some ideas from [ACN]. Paper V Sets with the Baire property in topologies formed from a given topology and 90 ideals of sets

In fact, let Oτ be a non-empty element of τ. Put Oσ = Intσ Oτ , where Intσ A (resp. Clσ A) is the interior (resp. closure) of A in the space (X, σ) for any A ⊆ X. Note that Oτ = Oσ ∪ ((Clσ Oσ \ Oσ) ∩ Oτ ) ∪ (Oτ \ Clσ Oσ). Note that the set (Clσ Oσ \ Oσ) ∩ Oτ is nowhere dense.

Claim 3.5. The set A = Oτ \ Clσ Oσ is nowhere dense.

Proof. Assume that the set A is not nowhere dense. Put Wσ = Intσ Clσ A. Note that Wσ 6= ∅, Wσ ⊆ Clσ A and Wσ ∩ Oσ = ∅. Since Wσ ∈ σ and τ is a π-network for σ, there exists an non-empty element Tτ of τ such that Tτ ⊆ Wσ. There are two possibilities:

Case 1. Tτ ∩ Oτ 6= ∅. Since Tτ ∩ Oτ ∈ τ and σ is a π-base for τ, there is a non-empty element Pσ of σ such that Pσ ⊆ Tτ ∩ Oτ . Note that Pσ ⊆ Oτ \ Oσ. We have a contradiction with the deﬁnition of Oσ.

Case 2. Tτ ∩ Oτ = ∅. Since σ is a π-network of τ, there exists a non-empty element Qσ of σ such that Qσ ⊆ Tτ . So Qσ ∩ Oτ = ∅ and hence Qσ ∩ Clσ Oτ = ∅. But Qσ ⊆ Wσ ⊆ Clσ A ⊆ Clσ Oτ . We have a contradiction. So A is nowhere dense.

By the use of the claim we get that Oτ is the union of Oσ and the nowhere dense set ((Clσ Oσ \ Oσ) ∩ Oτ ) ∪ (Oτ \ Clσ Oσ). We have done.

3.2 A special case of π-compatible topologies

Recall [CN] that the topology σ is said to be an admissible extension of τ if

(i) τ ⊆ σ, and

(ii) τ is a π-base for σ, i.e., for each non-empty element O of σ there is a non- empty element V of τ which is a subset of O.

Different examples of the admissible extensions of the usual topology on the reals one can find in [CH]: consider a subset A of R and define a topology τ(A) on R as follows:

(1) for each x ∈ A, {(x − ε, x + ε): ε > 0} is the neighborhood base at x.

(2) for each x ∈ R \ A, {[x, x + ε): ε > 0} is the neighborhood base at x.

Note that τ(∅) (resp. τ(R)) is the Sorgenfrey topology (resp. the usual topology) on the reals. All the topological spaces ((R, τ(A)),A ⊆ R, are regular hereditarily Lindelöf and hereditarily separable (see even [Kl] for the topological variety of the family of spaces). The following statement is trivial. 3 Two topologies on the same set and the Baire property 91

Proposition 3.6. Let X be a set, τ and σ be topologies on X such that σ is an admissible extension of τ. Then τ and σ are π-compatible.

Corollary 3.7. Let X be a set, τ and σ be topologies on X such that σ is an admissible extension of τ. Then the spaces (X, τ) and (X, σ) have the same families of nowhere dense sets (resp. meager sets or sets with the Baire property).

Proof. Apply Proposition 3.6, Lemma 3.3 and Theorem 3.4.

It is evident that the π-compatibility is an equivalence relation on the family of all topologies on a set X. So we have directly the following statement.

Corollary 3.8. Let X be a set, τ and σ be admissible extensions of some topology on X. Then τ and σ are π-compatible.

Remark 3.9. Since the examples from [CH] are admissible extensions of the usual topology on the reals, the families of nowhere dense sets (resp. meager sets or sets with the Baire property) are the same for all of them. Moreover, by Corollary 3.8 it follows that all topologies considered in [CH] are pairwise π-compatible. Let us also note that it is easy to suggest a pair τ, σ of topologies from [CH] such that σ * τ and τ * σ. We wonder if the inversion of Theorem 3.4 holds.

Question 3.10. Let X be a set, τ and σ be π-compatible topologies on X. Does there exist some topology ν on X such that τ and σ are admissible extensions of ν?

Now we will describe the admissible extensions of the usual topology on the reals via ideals of sets.

Proposition 3.11. Let (X, τ) be a topological space and I an ideal of sets on X ∗ such that I ⊆ In. Then the topology τ (I ) is an admissible extension of τ.

Proof. Since τ ⊆ τ ∗(I ), we need only to show that the topology τ is a π-base for the topology τ ∗(I ). Consider a non-empty element O of τ ∗(I ). We can assume that O = U \ I, where U ∈ τ and I ∈ I . Since I is a nowhere dense set in (X, τ), there exists a non-empty element W of τ such that W ⊆ U and W ∩ I = ∅. Note that W ⊆ O.

Let A ⊆ X. Denote by IA the ideal P(A) of all subsets of A.

1 Example 3.12. Let (R, τ) be the real line and A = { n : n = 1, 2,... } ⊆ R. Let us note that the set A is nowhere dense in the real line. Hence, IA ⊆ In. It ∗ follows from Proposition 3.11 that the topology τ (IA) is an admissible extension ∗ of τ. Since the set A is closed in the space (R, τ (IA)) and it is not closed in the ∗ ∗ real line, we have that τ 6= τ (IA). Moreover, the space (R, τ (IA)) is Hausdorﬀ but it is not regular. Paper V Sets with the Baire property in topologies formed from a given topology and 92 ideals of sets

Proposition 3.13. Let (X, τ) be a T1-space without isolated points, I an ideal of ∗ sets on X and the topology τ (I ) be an admissible extension of τ. Then I ⊆ In.

Proof. Assume that there exists an element I of I such that O = Int Cl(I) 6= ∅. Put U = O \ (O ∩ I) if O \ (O ∩ I) 6= ∅ or U = {p}, where p ∈ O, otherwise. Note that U 6= ∅, U ∈ τ ∗(I ) and there exists no non-empty element W of τ such that W ⊆ U. We have a contradiction.

Remark 3.14. The condition on the space (X, τ) "to be without isolated points" in Proposition 3.13 is essential. Indeed, let (X, τ) be a discrete space and I = P(X). Note that τ ∗(I ) = τ, i.e., τ ∗(I ) is an admissible extension τ, and In = {∅}.

Theorem 3.15. Let (X, τ) be a T1-space without isolated points (for example, the real line) and I an ideal of sets on X. Then the topology τ ∗(I ) is an admissible extension of τ iﬀ I ⊆ In.

Proof. Apply Propositions 3.11 and 3.13.

∗ Remark 3.16. Note that the topology τ (IA) from Example 3.12 as well as ∗ 1 the topology τ (IB), where B = {2 + n : n = 1, 2,... } ⊆ R, is an admissible ∗ ∗ extension of the usual topology τ on the reals, and τ (IA) * τ (IB) as well as ∗ ∗ ∗ ∗ τ (IB) * τ (IA). However, the topologies τ (IA), τ (IB) are π-compatible.

Remark 3.17. Note that the Sorgenfrey topology is not a ∗-topology of the usual topology on the reals.

3.3 Codense ideals of sets and sets with the Baire property

Theorem 3.18. Let (X, τ) be a topological space and I an ideal of sets on X such that τ ∩ I = {∅}. Then for any subset N of X the following holds. If N is a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, τ) then N is also a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, τ ∗(I )).

Proof. Let N be a nowhere dense set in the space (X, τ). Consider a non-empty element O of τ ∗(I ). We can assume that O = U \ I, where U ∈ τ and I ∈ I . Since U 6= ∅, there exists a non-empty element W of τ such that W ⊆ U and W ∩ N = ∅. Note that W \ I 6= ∅, W \ I ⊆ O and (W \ I) ∩ N = ∅. Hence, N is a nowhere dense set in the space (X, τ ∗(I )). The case of meager sets is trivial. ∗ ∗ Since τ ⊆ τ (I ), it is easy now to see that Bp(X, τ) ⊆ Bp(X, τ (I )).

∗ ∗ Let In (resp. Im) denote the family of nowhere dense (resp. meager) sets in the space (X, τ ∗(I )). 3 Two topologies on the same set and the Baire property 93

Lemma 3.19. Let (X, τ) be a topological space and I an ideal of sets on X such ∗ that τ ∩ I = {∅}. Then for each I ∈ I we have I ∈ In .

Proof. Let U be a non-empty element of τ ∗(I ). We can assume that U = O \ J, where O ∈ τ and J ∈ I . Note that I ∪J ∈ I and V = O \(I ∪J) ∈ τ ∗(I ). Since τ ∩ I = {∅}, we have even that V 6= ∅. Moreover, V ⊆ U and V ∩ I = ∅.

Example 3.20. Let (R, τ) be the real line and V a Vitali set there. Let us note that no Vitali set contains a non-empty open set of the real line. So τ ∩IV = {∅}. It follows from Theorem 3.18 that every nowhere dense set (resp. a meager set or a set with the Baire property) in the space (R, τ) is also a nowhere dense set ∗ (resp. a meager set or a set with the Baire property) in the space (R, τ (IV )). However we have no equality between the corresponding families. In fact, observe ∗ that the set V is nowhere dense in the space (R, τ (IV )) (see Lemma 3.19) and it does not have the Baire property in the real line.

Remark 3.21. It is easy to see that without the condition τ ∩ I = {∅} in ∗ Theorem 3.18 the inclusion In ⊆ In does not valid. Indeed, let (R, τ) be the real ∗ ∗ line and I any ideal of sets such that τ (I ) = τdisc. Note that In = {∅} and ∗ In 6= {∅}. But Bp(R, τ) ⊆ Bp(R, τ (I )) = P(R).

Corollary 3.22. Let (X, τ) be a topological space and I an ideal of sets on X ∗ such that τ ∩ I = {∅} and In = In. Then I ⊆ In (and hence by Proposition 3.11, τ ∗(I ) is an admissible extension of τ).

∗ ∗ Proof. By Lemma 3.19 we have I ⊆ In . Since In = In, we get that I ⊆ In.

Let us observe that in general the condition τ ⊆ σ does not need to imply even the inclusion of the family of nowhere dense sets in the space (X, τ) to the family of sets with the Baire property in the space (X, σ).

Example 3.23. Let (R, τ) be the real line and the topology τ0 (resp. τ1) be generated by the open intervals (−∞, a), a ∈ R (resp. the complements to all compact subsets of the real line). It is easy to see that for each i = 0, 1 we have τi ⊆ τ and every bounded Vitali set (which does not belong to Bp(R, τ)) is a nowhere dense subset of the space (R, τi). Let us note that the space (R, τ0) (resp. (R, τ1)) is a T0-(resp. T1-)space. Is there a T2-space of this type?

Lemma 3.24 ([CJ]). Let (X, τ) be a second countable topological space and I a σ-ideal of sets on X such that τ ∩ I = {∅}. Then a subset N of X is an element ∗ of In iﬀ N = P ∪ I where P ∈ In and I ∈ I .

Corollary 3.25. Let (X, τ) be a second countable topological space and I a σ- ideal of sets on X such that τ ∩ I = {∅}. Then a subset M of X is an element ∗ of Im iﬀ M = Q ∪ I where Q ∈ Im and I ∈ I . Paper V Sets with the Baire property in topologies formed from a given topology and 94 ideals of sets

∗ S∞ ∗ Proof. Let M ∈ Im. So M = i=1 Ni, where Ni ∈ In for each i. By Lemma 3.24, Ni = Pi ∪ Ii, where Pi ∈ In and Ii ∈ I for each i. So we have M = S∞ S∞ S∞ S∞ S∞ i=1(Pi ∪ Ii) = ( i=1 Pi) ∪ ( i=1 Ii). Put Q = i=1 Pi and I = i=1 Ii. It is clear that Q ∈ Im and I ∈ I . S∞ Let now M = Q ∪ I, where Q ∈ Im and I ∈ I . So Q = i=1 Pi, where S∞ S∞ Pi ∈ In for each i, and M = ( i=1 Pi) ∪ I = i=1(Pi ∪ Ii), where I1 = I and ∗ Ii = ∅ for i ≥ 2. Since Pi ∪ Ii ∈ In for each i (see again Lemma 3.24), it follows ∗ that M ∈ Im.

Recall [Ha] that a topology τ on a set X is said to be compatible with an ideal I of sets on X, denoted by τ ∼ I , if the following holds for every A ⊆ X: if for every x ∈ A there exists a U ∈ N (x) such that U ∩ A ∈ I , then A ∈ I . Recall (cf. [JH]) that

(i) If τ ∼ I then β(τ, I ) = τ ∗(I ).

(ii) if (X, τ) is a hereditarily Lindelöf space (in particular, a second countable topological space) and I is a σ-ideal of sets on X then τ ∼ I .

Theorem 3.26. Let (X, τ) be a second countable topological space and I a σ- ideal of sets on X such that τ ∩ I = {∅}. If I ⊆ Im then the spaces (X, τ) and (X, τ ∗(I )) have the same families of meager sets (resp. sets with the Baire property).

∗ Proof. By Theorem 3.18 in particular we have that Im ⊆ Im and Bp(X, τ) ⊆ ∗ Bp(X, τ (I )). ∗ ∗ Let us show that Im ⊇ Im. In fact, if M ∈ Im then by Corollary 3.25 we have that M = Q ∪ I, where Q ∈ Im and I ∈ I . Since I ⊆ Im (by the assumption), we get that M ∈ Im. ∗ Let us show the inclusion Bp(X, τ) ⊇ Bp(X, τ (I )). Suppose that A ∈ ∗ ∗ ∗ Bp(X, τ (I )), that is A = (O \ E1) ∪ E2, where O ∈ τ (I ) and E1,E2 ∈ Im. Since τ ∼ I (see (ii) and (i) above), it follows that O = U \ I, where U ∈ τ and I ∈ I . So A = (U \(I ∪E1))∪E2. Let us note that by Corollary 3.25, E1 = Q1 ∪I1 and E2 = Q2 ∪ I2, where Q1,Q2 ∈ Im and I1,I2 ∈ I . Since I ⊆ Im (by the assumption), we have I ∪ E1,E2 ∈ Im which implies that A ∈ Bp(X, τ). Hence, ∗ Bp(X, τ) ⊇ Bp(X, τ (I )).

Example 3.27. Let (R, τ) be the real line and Q the set of rational numbers.

It is evident that IQ is a σ-ideal of sets on R, τ ∩ IQ = {∅} and IQ ⊆ Im. It ∗ follows from Theorem 3.26 that the spaces (R, τ) and (R, τ (IQ)) have the same families of meager sets (resp. sets with the Baire property). However the families ∗ of nowhere dense sets in the spaces (R, τ) and (R, τ (IQ)) are not equal. Indeed the set Q is everywhere dense in the space (R, τ) but it is nowhere dense in the ∗ space (R, τ (IQ)) (see Lemma 3.19). 3 Two topologies on the same set and the Baire property 95

3.4 Partial equality of the families of nowhere dense sets (resp. meager sets or sets with the Baire property)

We have considered above a suﬃcient condition on two topologies on a set to produce the same families of nowhere dense sets (resp. meager sets or sets with the Baire property). Now we will suggest a suﬃcient condition on two topologies to produce partially the same families of nowhere dense sets (resp. meager sets or sets with the Baire property).

Let τ be a topology on a set X and Y ⊆ X. Denote by τY the topology {U ∩ Y : U ∈ τ} on the set Y . The following statement is evident.

Proposition 3.28. Let A ⊆ Y ⊆ X and τ be a topology on X.

(i) If A is a nowhere dense (resp. meager) set in (Y, τY ) then A is a nowhere dense (resp. meager) set in (X, τ). (The inversion is not valid in general.)

(ii) If Y ∈ τ then A is a nowhere dense set (resp. a meager set or a set with the

Baire property) in (Y, τY ) iﬀ A is a nowhere dense set (resp. a meager set or a set with the Baire property) in (X, τ).

Theorem 3.29. Let τ, σ be topologies on a set X and Y a non-empty subset of X such that Y ∈ τ ∩ σ. Let also τY and σY be π-compatible topologies on the set Y . Then for any subset A of Y we have: A is a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, τ) iﬀ A is a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, σ).

Proof. Apply Theorem 3.4 and Proposition 3.28 (ii).

Corollary 3.30. Let X be a set and τ, σ topologies on X such that τ ⊆ σ. Let also there exists a non-empty subset Y of X such that Y ∈ τ and for every U ∈ σ we have Y ∩ U ∈ τ. Then for any subset A of Y we have: A is a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, τ) iﬀ A is a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, σ).

Proof. Note that Y ∈ σ and τY , σY coincide. Then apply Theorem 3.29.

Corollary 3.31. Let (X, τ) be a topological space, I be an ideal of sets on X such that Y = X \ Cl(S{I : I ∈ I }) 6= ∅. Then for any subset A of Y we have: A is a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, τ) iﬀ A is a nowhere dense set (resp. a meager set or a set with the Baire property) in the space (X, τ ∗(I )). Paper V Sets with the Baire property in topologies formed from a given topology and 96 ideals of sets

Proof. Since τ ⊆ τ ∗(I ) and Y ∈ τ, we need only to show (see Corollary 3.30) that for each U ∈ τ ∗(I ), U ∩ Y ∈ τ. Recall that the family β(τ, I ) = {W \ I : ∗ S W ∈ τ, I ∈ I } is a base for the topology τ (I ). So U = {Wα \ Iα : Wα ∈ S τ, Iα ∈ I , α ∈ A}. Hence, U ∩ Y = ( {Wα \ Iα : Wα ∈ τ, Iα ∈ I , α ∈ A}) ∩ Y = S {Wα ∩ Y : Wα ∈ τ, α ∈ A} ∈ τ.

Example 3.32. Let (R, τ) be the real line, A = (0, 2) and B = (3, 5) subsets of R. Consider Vitali sets VA,VB such that VA ⊂ A and VB ⊂ B. Note that none of the families of nowhere dense sets (resp. meager sets or sets with the Baire ∗ ∗ property) related to the spaces (R, τ (IA)), (R, τ (IB)) belongs to other. Indeed, ∗ it is easy to see that each subset of A (resp. B) is clopen in the space (R, τ (IA)) ∗ ∗ (resp. (R, τ (IB))). Hence, the sets {1},VA are elements of τ (IA) and the sets ∗ {4},VB are elements of τ (IB). In particular, the set {1} (resp. {4}) is not mea- ∗ ∗ ∗ ger in the space (R, τ (IA)) (resp. (R, τ (IB))) and VA ∈ Bp(R, τ (IA)) (resp. ∗ VB ∈ Bp(R, τ (IB))). Since the sets {1}, {4} are nowhere dense in the real line and the sets VA,VB do not belong to the family Bp(R, τ), it follows from Corollary ∗ 3.31 that the set {1} (resp. {4}) is nowhere dense in the space (R, τ (IB)) (resp. ∗ ∗ ∗ (R, τ (IA))) and VA ∈/ Bp(R, τ (IB)) (resp. VB ∈/ Bp(R, τ (IA))). Let us also ∗ ∗ note that both spaces (R, τ (IA)), (R, τ (IB)) are homeomorphic to the topological union of two copies of half-open interval (0, 1] and one copy of the discrete space of cardinality continuum.

4 Three and more topologies on the same set and the Baire property

4.1 Ideals of sets producing equal ∗-topologies

Lemma 4.1. Let (X, τ) be a space and I an ideal of sets on X. Then the following statements are equivalent.

(i) τ = τ ∗(I ).

(ii) Every element of I is a closed set in the space (X, τ).

(iii) Every element of I is a closed and discrete set in the space (X, τ).

Proof. Recall that every element I of I is closed and discrete in the topological space (X, τ ∗(I )). So if τ = τ ∗(I ) then every element I of I is closed and discrete in the space (X, τ). Recall also that the family β(τ, I ) = {O \ I : O ∈ τ, I ∈ I } is a base for the topology τ ∗(I ). Hence, if every element of I is a closed set in the space (X, τ) then the family β(τ, I ) consists of open sets of the space (X, τ) what implies that τ = τ ∗(I ). 4 Three and more topologies on the same set and the Baire property 97

Denote by Icd the ideal of all closed and discrete subsets of the space (X, τ).

Proposition 4.2. Let (X, τ) be a space. Then for any ideal I of sets on X such ∗ ∗ that I ⊆ Icd we have τ = τ (I ) = τ (Icd).

∗ ∗ Proof. By Lemma 4.1 we have τ (I ) = τ = τ (Icd).

Example 4.3. Let (R, τ) be the real line, IZ the ideal of integers and If the ideal of ﬁnite subsets of R. Note that If ( Icd and IZ ( Icd. Moreover, If \ IZ 6= ∅ ∗ ∗ ∗ and IZ \ If 6= ∅. By Proposition 4.2 we have τ = τ (IZ) = τ (If ) = τ (Icd).

Proposition 4.4. Let (X, τ) be a topological space and let I1 and I2 be ideals of sets on X. If I1 ⊆ I2 and for each J ∈ I2 there exist a closed set F in (X, τ) ∗ ∗ and an element I ∈ I1 such that J = F ∪ I then τ (I1) = τ (I2).

∗ ∗ Proof. It is enough to show that τ (I1) and τ (I2) have the same base. It is clear that β(τ, I1) ⊆ β(τ, I2). Consider A ∈ β(τ, I2). Recall that A = O \ J, where O ∈ τ and J ∈ I2. Keeping in mind that J = F ∪ I, where F is closed in (X, τ) and I ∈ I1, we get that A = O \ (F ∪ I) = (O \ F ) \ I. Since O \ F ∈ τ, we have that A ∈ β(τ, I1). Hence, β(τ, I2) ⊆ β(τ, I1) from which it follows that ∗ ∗ β(τ, I2) = β(τ, I1). So τ (I1) = τ (I2).

Example 4.5. Let (R, τ) be the real line, A = (0, 2) and B = [0, 2] subsets of R. Note that IA ( IB and for each J ∈ IB there exist a closed set F (either ∅, {0}, {2} or {0} ∪ {2}) in (R, τ) and an element I ∈ IA such that ∗ ∗ J = F ∪ I. By Proposition 4.4 we get τ (IA) = τ (IB). In particular, we ∗ ∗ ∗ have Bp(R, τ (IA)) = Bp(R, τ (IB)). Note that Bp(R, τ (IA)) 6= Bp(R, τ) ∗ and Bp(R, τ (IA)) 6= P(R) (see Example 3.32).

Proposition 4.6. Let (X, τ) be a topological space and let I1 and I2 be ideals ∗ ∗ of sets on X such that I1 ⊆ I2. If A (I1) ⊆ A (I2) for every A ⊆ X then ∗ ∗ τ (I1) = τ (I2).

∗ ∗ ∗ ∗ Proof. Clearly, if A (I1) ⊆ A (I2) then A (I1) = A (I2) (see Theorem 2.2 (v)). ∗ ∗ ∗ ∗ So A ∪ A (I1) = A ∪ A (I2) for every A ⊆ X and hence τ (I1) = τ (I2).

Remark 4.7. Note that the described in Proposition 4.6 case with the inclusion ∗ ∗ A (I1) ⊆ A (I2) for every A ⊆ X can occur. Indeed, let (R, τ) be the real line and Ik = {A ⊆ R : Cl(A) is a bounded subset of R}. Let A ⊆ R. Since for each ∗ x ∈ R we can ﬁnd U ∈ N (x) such that A ∩ U ∈ Ik then A (Ik) = ∅. So, ∗ ∗ ∗ ∗ A (Ik) = A (P(R)) = ∅ for each A ⊆ R. Hence τ (Ik) = τ (P(R)) = τdisc.

Recall that an open subset U of a topological space (X, τ) is said to be regular open if U = Int Cl(U). The family of all regular open sets in (X, τ) forms a base for a new topology on X known as the semiregularization of τ, denoted by τs, with the property that τs ⊆ τ. The topology τ is said to be semiregular if τ = τs. Paper V Sets with the Baire property in topologies formed from a given topology and 98 ideals of sets

Let us note that if τ is a regular topological space (i.e. for every non-empty closed set F and every point x∈ / F there exist disjoint open sets OF and Ox such that F ⊆ OF and x ∈ Ox) then τ is semiregular.

Theorem 4.8 ([JH]). Let (X, τ) be a topological space with an ideal of sets I on X such that τ ∩ I = {∅}. Then we have the following:

∗ (i) τs = (τ (I ))s.

(ii) If τ ∗(I ) is semiregular then τ = τ ∗(I ).

Proposition 4.9. Let (X, τ) be a topological space and let I1, I2 be ideals of sets on X such that τ ∩ Ii = {∅} for i = 1, 2. Then we have the following:

∗ ∗ (i) (τ (I1))s = (τ (I2))s.

∗ ∗ ∗ (ii) If I1 ⊆ I2 and τ (I2) is semiregular then τ (I1) = τ (I2).

∗ Proof. (i) In the virtue of Theorem 4.8 (i) we have τs = (τ (Ii))s for i = 1, 2 and ∗ ∗ therefore (τ (I1))s = (τ (I2))s. ∗ (ii) Since I1 ⊆ I2 and τ (I2) is semiregular (by the assumption), it follows ∗ ∗ ∗ ∗ ∗ from (i) that τ (I2) = (τ (I2))s = (τ (I1))s ⊆ τ (I1) ⊆ τ (I2) and thus ∗ ∗ τ (I1) = τ (I2).

4.2 Ideals of sets producing diﬀerent ∗-topologies

For A and B in P(X) deﬁne A ∨ B = {A ∪ B : A ∈ A ,B ∈ B}. It is evident that if A and B are ideals of sets on X then A ∨ B is also an ideal of sets on X and if A ⊆ B then A ∨ B = B.

Proposition 4.10. Let (X, τ) be a topological space and I , J be ideals of sets on ∗ X such that τ ∩ (I ∨ J ) = {∅} and J ⊆ In. Then τ (I ∨ J ) is an admissible extension of τ ∗(I ) on X.

Proof. Since τ ∗(I ) ⊆ τ ∗(I ∨ J ), we need only to show that the topology τ ∗(I ) is a π-base for the topology τ ∗(I ∨ J ). Consider a non-empty element O of ∗ τ (I ∨ J ). We can assume that O = U \ (I1 ∪ I2), where U ∈ τ, I1 ∈ I and I2 ∈ J . Since I2 is a nowhere dense set in (X, τ), there exists a non-empty element W of τ such that W ⊆ U and W ∩ I2 = ∅. Since τ ∩ I = {∅}, we have ∗ W \ I1 6= ∅. Note that W \ I1 ⊆ O and W \ I1 ∈ τ (I ).

Example 4.11. Let (R, τ) be the real line and A = Q ∩ (1, 2). It is easy to ∗ ∗ see that (IA ∨ In) ∩ τ = {∅}. Moreover, τ (IA) 6= τ (IA ∨ In). In fact, let 1 ∗ B = { n : n = 1, 2,... }. Note that the set B is closed in the space (R, τ (IA ∨In)) ∗ but B is not closed in the space (R, τ (IA)). 4 Three and more topologies on the same set and the Baire property 99

Proposition 4.12. Let (X, τ) be a topological space and let I1, I2 be ideals of sets on X such that I1 ⊆ I2 and τ ∩ I2 = {∅}. Then for any subset N of X the following holds. If N is a nowhere dense set (resp. a meager set or a set with the ∗ Baire property) in the space (X, τ (I1)) then N is also a nowhere dense set (resp. ∗ a meager set or a set with the Baire property) in the space (X, τ (I2)).

∗ Proof. Let N be a nowhere dense set in the space (X, τ (I1)). Consider a non- ∗ empty element O of τ (I2). We can assume that O = U \ I2, where U ∈ τ and ∗ I2 ∈ I2. Since U 6= ∅ and U ∈ τ (I1), there exists W ∈ τ and I1 ∈ I1 such that W \ I1 6= ∅, W \ I1 ⊆ U and (W \ I1) ∩ N = ∅. Note that I = I1 ∪ I2 ∈ I2, W \ I 6= ∅, W \ I ⊆ O and (W \ I) ∩ N = ∅. Hence, N is a nowhere dense set in ∗ the space (X, τ (I2)). ∗ ∗ The case of meager sets is trivial. Since τ (I1) ⊆ τ (I2), it is easy now to see ∗ ∗ that Bp(X, τ (I1)) ⊆ Bp(X, τ (I2)).

Example 4.13. Let (R, τ) be the real line and V a Vitali subset of R. Consider the following ideals of sets I1 = {F ⊆ V : F is ﬁnite} and I2 = P(V ). It is clear that I1 ⊆ I2 and τ ∩ I2 = {∅}. It follows by Proposition 4.12 that any nowhere dense set (resp. a meager set or a set with the Baire property) in the ∗ space (X, τ (I1)) is also a nowhere dense set (resp. a meager set or a set with the ∗ Baire property) in the space (X, τ (I2)). However, we have no equality between ∗ the corresponding families. In fact, by Proposition 4.2, τ (I1) = τ. Then check Example 3.20.

Theorem 4.14. Let (X, τ) be a second countable topological space and I , J σ- ideals of sets on X such that τ ∩ I = τ ∩ J = {∅}. If I , J ⊆ Im then the spaces (X, τ ∗(I )) and (X, τ ∗(J )) have the same families of meager sets (resp. sets with the Baire property).

Proof. Apply Theorem 3.26.

Example 4.15. Let (R, τ) be the real line and A = Q ∩ (0, 1), B = Q ∩ (2, 3). It is evident that IA, IB are σ-ideals of sets on R, τ ∩ IA = τ ∩ IB = {∅} ∗ and IA, IB ⊆ Im. It follows from Theorem 4.14 that the spaces (R, τ (IA)) ∗ and (R, τ (IB)) have the same families of meager sets (resp. sets with the Baire ∗ property). However the families of nowhere dense sets in the spaces (R, τ (IA)) ∗ and (R, τ (IB)) are not equal. Indeed the set A (resp. B) is nowhere dense in the ∗ ∗ space (R, τ (IA)) (resp. (R, τ (IB))) but it is not nowhere dense in the space ∗ ∗ (R, τ (IB)) (resp. (R, τ (IA))). Paper V Sets with the Baire property in topologies formed from a given topology and 100 ideals of sets 5 Applications of the local function for proving some known facts

In this section we show how the local function can be used to prove some known facts. We shall be mostly interested in facts which are related to Bernstein sets and Vitali sets of the real line. Let us recall the following statement.

Theorem 5.1 (cf. [Va]). Let (X, τ) be a topological space and A ⊆ X. Then

∗ (i) A \ Int(A (Im)) is meager;

∗ (ii) A ∈ Bp(X, τ) iﬀ Int(A (Im)) \ A is meager.

The next statement shows that the sets having the Baire property with the same local function, are equal up to a meager set.

Proposition 5.2. Let (X, τ) be a topological space and A, B ⊆ X. If A, B ∈ ∗ ∗ Bp(X, τ) and A (Im) = B (Im) then A∆B is meager.

Proof. Since A ∈ Bp(X, τ) and B ∈ Bp(X, τ), then by Theorem 5.1 (ii) we have ∗ ∗ Int(A (Im)) \ A ∈ Im and Int(B (Im)) \ B ∈ Im. It follows also from The- ∗ ∗ orem 5.1 (i) that A \ Int(A (Im)) ∈ Im and B \ Int(B (Im)) ∈ Im. So ∗ ∗ ∗ ∗ A∆ Int(A (Im)) ∈ Im and Int(B (Im))∆B ∈ Im. Since A (Im) = B (Im), ∗ ∗ it follows that A∆B = (A∆ Int(A (Im))∆(Int(B (Im))∆B) and hence A∆B is meager.

5.1 Applications to Bernstein sets

Let us recall that a subset B of the real line is called a Bernstein set if F ∩ B 6= ∅ and F ∩ (R \ B) 6= ∅ for every uncountable closed set F ⊆ R. Note that R \ B is also a Bernstein set and none of the sets B and R \ B is meager in (R, τ).

Lemma 5.3. Let (R, τ) be the real line and B a Bernstein subset of R. Then ∗ ∗ B (Im) = (R \ B) (Im) = R.

Proof. Suppose that B is a Bernstein subset of R. For a non-meager subset Y 00 of R, let OY = Int Cl(Y ) and Y = {x ∈ Y ∩ OY : Y ∩ U/∈ Im for every U ∈ (x)} O 00 = O 00 = N . It is known (see Lemma 4.1 in [ACN]) that B (R\B) R. Since 00 ∗ ∗ B ⊆ B (Im) and B (Im) is closed in (R, τ) (see Theorem 2.2 (iii)), we have ∗ ∗ ∗ that OB00 ⊆ B (Im) and thereby B (Im) = R. Similarly, (R\B) (Im) = R.

Proposition 5.4 (cf. [Ox]). If B is a Bernstein subset of R then B/∈ Bp(R, τ). 5 Applications of the local function for proving some known facts 101

∗ ∗ Proof. Suppose that B ∈ Bp(R, τ). Since B (Im) = R (Im) = R and R ∈ Bp(R, τ), it follows from Proposition 5.2 that R∆B = R \ B ∈ Im. But this is a contradiction.

Proposition 5.5. Let (X, τ) be a topological space and A ⊆ B ⊆ X. Then we have the following.

∗ ∗ (i) If A ∈ Bp(X, τ) and A (Im) = B (Im) then B ∈ Bp(X, τ).

∗ (ii) If A ∈ Bp(X, τ) and B ⊆ Cl (A) then B ∈ Bp(X, τ).

∗ Proof. (i) Since A ∈ Bp(X, τ), we have Int(A (Im))\A ∈ Im. By the assumption ∗ ∗ ∗ ∗ we have that A ⊆ B and A (Im) = B (Im). So Int(B (Im))\B = Int(A (Im))\ ∗ B ⊆ Int(A (Im)) \ A ∈ Im. It follows from Theorem 5.1 (ii) that B ∈ Bp(X, τ). ∗ ∗ (ii) From the inclusions A ⊆ B ⊆ Cl (A) = A ∪ A (Im) it follows that ∗ ∗ ∗ ∗ ∗ ∗ ∗ A (Im) ⊆ B (Im) ⊆ [A ∪ A (Im)] (Im) = A (Im). Since A (Im) = B (Im), we get the statement from (i).

Theorem 5.6 (cf. [Ku]). Let (X, τ) be a topological space and A ⊆ X. Then ∗ ∗ A ∈ Bp(X, τ) iﬀ A (Im) ∩ (X \ A) (Im) is a nowhere dense set.

Now, it is easy to show that every subset of a Bernstein set with the Baire property is meager.

Proposition 5.7 (cf. [Ox]). Let (R, τ) be the real line and B a Bernstein subset of R. If A is a subset of B such that A ∈ Bp(R, τ), then A is a meager set in (R, τ).

∗ ∗ Proof. Suppose that A ⊆ B and A ∈ Bp(R, τ). Then A (Im)∩(R\A) (Im) ∈ In ∗ ∗ by Theorem 5.6. Since (R \ A) (Im) ⊇ (R \ B) (Im) = R (see Lemma 5.3), we ∗ ∗ ∗ have A (Im) ∈ In. Note that A = (A\A (Im))∪(A∩A (Im)) and by Theorem ∗ 5.1 (i) we have A \ A (Im) ∈ Im. So A ∈ Im.

5.2 Applications to Vitali sets

Let V be the family of all Vitali sets on R, and let SV be the family of all ﬁnite Sn unions of elements of V, i.e., SV = { i=1 Vi : Vi ∈ V, n ∈ N}. Recall that each Vitali set is non-meager on the real line, that is, V ∩ Im = ∅. For A and B in P(R) deﬁne A ∆B = {A∆B : A ∈ A ,B ∈ B}. Note that Bp(R, τ) = τ∆Im. It was proved in [CN] that the family SV ∆Im consists of sets without the Baire property in (R, τ). Here we will show this fact with the help of local function.

Lemma 5.8 ([Ch]). For any set U ∈ SV and any non-empty open set O of R, there is a V ∈ V such that V ⊂ O \ U. Paper V Sets with the Baire property in topologies formed from a given topology and 102 ideals of sets

By this lemma it is easy to obtain the local function of the set R \ U with the respect to Im for each U ∈ SV .

Proposition 5.9. Let (R, τ) be the real line and let U be any element of SV . Then ∗ we have (R \ U) (Im) = R.

∗ Proof. Suppose that U ∈ SV . Then (R \ U) (Im) = {x ∈ R : O ∩ (R \ U) ∈/ Im for every O ∈ N (x)} = {x ∈ R : O \ U/∈ Im for every O ∈ N (x)}. By Lemma 5.8 there is V ∈ V such that V ⊆ O \ U. Since V ∩ Im = ∅ then ∗ O \ U/∈ Im for every O ∈ N (x). Hence, (R \ U) (Im) = R.

Proposition 5.10 ([Ch]). Let (R, τ) be the real line and U any element of SV . Then R \ U/∈ Bp(R, τ) and U/∈ Bp(R, τ).

Proof. Suppose that R\U ∈ Bp(R, τ) for some U ∈ SV . It follows from Proposition ∗ 5.9 and Theorem 5.1 (ii) that Int((R \ U) (Im)) \ (R \ U) = Int(R) \ (R \ U) = R \ (R \ U) = U ∈ Im. This is a contradiction. So R \ U/∈ Bp(R, τ). Since Bp(R, τ) is a σ-algebra of sets on R, it follows that U/∈ Bp(R, τ).

Theorem 5.11 ([CN]). Let (R, τ) be the real line. If A ∈ SV ∆Im then A/∈ Bp(R, τ).

Proof. Assume that there exists A ∈ (SV ∆Im) ∩ (Bp(R, τ)). Then A = U∆F ∗ ∗ for some U ∈ SV and F ∈ Im,A (Im) = U (Im) (see Theorem 2.2) and ∗ ∗ Int(A (Im)) \ A ∈ Im (see Theorem 5.1 (ii)). Moreover, Int(A (Im)) \ A = ∗ ∗ ∗ Int(A (Im)) \ (U∆F ) ⊇ Int(A (Im)) \ (U ∪ F ) = (Int(U (Im)) \ U) \ F .

∗ Claim 5.12. The set M = (Int(U (Im)) \ U) \ F is not meager on the real line (R, τ).

∗ Proof. Assume that M ∈ Im. The set Int(U (Im))\U can be written in the form ∗ ∗ ∗ Int(U (Im))\U = ((Int(U (Im)) \ U) ∩ F )∪M and therefore Int(U (Im))\U ∈ Im. Thus by Theorem 5.1 (ii), U ∈ Bp(R, τ). We have a contradiction (see Proposition 5.10 ) which proves the claim.

∗ It follows from the Claim 5.12 that Int(A (Im)) \ A/∈ Im and hence by Theorem 5.1 (ii), A/∈ Bp(R, τ). This is a contradiction with our assumption which proves the theorem.

∗ Corollary 5.13. Let (R, τ) be the real line. If A ∈ SV ∆Im then A/∈ Bp(R, τ (Im)).

∗ Proof. By Proposition 3.26 we have Bp(R, τ) = Bp(R, τ (Im)). So the statement follows from Theorem 5.11. References 103

References

[ACN] M. Aigner, V. A Chatyrko and V. Nyagahakwa, On countable families of sets without the Baire property, Colloq. Math., 2013, Vol.133, No.2, 179 - 187.

[Ch] V. A. Chatyrko, On Vitali sets and their unions, Matematicki. Vesnik, 2011, Vol.63, No.2, 87-92.

[CH] V. A. Chatyrko and Y. Hattori, A poset of topologies on the set of real numbers, Comment. Math. Univ. Carolin., 2013, Vol.54, No.2, 189 - 196.

[CN] V. A. Chatyrko and V. Nyagahakwa, On the families of sets without the Baire property generated by Vitali sets, p-Adic Numbers and Ultrametric Anal. Appl., 2011, Vol.3, No.2, 100 - 107.

[CJ] K. Ciesielski and J. Jasinski, Topologies making a given ideal nowhere dense or meager, Topol. Appl., 1995, Vol.63, No.3, 277 - 298.

[DGR] J. Dontchev, M. Ganster, and D. Rose, Ideal resolvability, Topol. Appl., 1999, Vol.93, No.1, 1-16.

[JH] D. Janković and T.R. Hamlett, New topologies from old via Ideals, Amer. Math. Monthly, 1990, Vol.97, No.4, 295 - 310.

[Ha] E. Hayashi, Topologies deﬁned by local properties, Math. Ann., 1964, Vol.156, No.3, 205 - 215.

[Kl] J. Kulesza, Some results on spaces between the Sorgenfrey topology and the usual topology on the reals, manuscript, 2015.

[Ku] K. Kuratowski, Topology, Volume I, PWN-Polish Scientiﬁc Publishers and Academic Press, Warszaw and New York, 1966.

[Mo] S. Modak, Some new topologies on ideal topological Spaces, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2012, Vol.82, No.3, 233 - 244.

[Ox] J.C. Oxtoby, Measure and Category, Graduate Texts in Mathematics: A Survey of the analogies between topological and measure spaces, Springer- Verlag, New York, Heldenberg, Berlin, 1971.

[Va] M. Valdivia, Topics in locally convex spaces, North Holland Publishing Com- pany, 1982.

[Vi] G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, 1905.