Linköping Studies in Science and Technology.
Thesis No. 1700
Semigroups of Sets Without the
Baire Property In Finite Dimensional
Euclidean Spaces
Vénuste NYAGAHAKWA
Department of Mathematics
Division of Mathematics and Applied Mathematics
Linköping University, SE–581 83 Linköping, Sweden
Linköping Studies in Science and Technology. Thesis No. 1700
Semigroups of Sets Without the Baire Property In Finite Dimensional Euclidean Spaces
Vénuste NYAGAHAKWA venuste.nyagahakwa@liu.se
www.mai.liu.se
Mathematics and Applied Mathematics Department of Mathematics
Linköping University SE–581 83 Linköping
Sweden
ISBN 978-91-7519-146-1 ISSN 0280-7971 Copyright © 2015 Vénuste NYAGAHAKWA
To Iribagiza M. Rose and Mizero B. Roberto.
Abstract
A semigroup of sets is a family of sets closed under finite unions. This thesis fo-cuses on the search of semigroups of sets in finite dimensional Euclidean spaces Rn, n ≥1, which elements do not possess the Baire property, and on the study
of their properties.
Recall that the family of sets having the Baire property in the real lineR, is a
σ−algebra of sets, which includes both meager and open subsets ofR. However,
there are subsets ofR which do not belong to the algebra. For example, each classical Vitali set onR does not have the Baire property.
It has been shown by Chatyrko that the family of all finite unions of Vitali sets on the real line, as well as its natural extensions by the collection of meager sets, are (invariant under translations ofR) semigroups of sets which elements do not possess the Baire property.
Using analogues of Vitali sets, when the group Q of rationals in the Vitali construction is replaced by any countable dense subgroup Q of reals, (we call the sets Vitali Q-selectors ofR) and Chatyrko’s method, we produce semigroups of sets onR related to Q, which consist of sets without the Baire property and which are invariant under translations of R. Furthermore, we study the rela-tionship in the sense of inclusion between the semigroups related to different Q. From here, we define a supersemigroup of sets based on all Vitali selectors of R. The defined supersemigroup also consists of sets without the Baire property and is invariant under translations of R. Then we extend and generalize the results from the real line to the finite-dimensional Euclidean spacesRn, n≥ 2, and indicate the difference between the cases n=1 and n≥2. Additionally, we show how the covering dimension can be used in defining diverse semigroups of sets without the Baire property.
Populärvetenskaplig sammanfattning
En semigrupp av mängder är en familj av mängder som är sluten under ändliga unioner. Denna licentiatavhandling fokuserar på sökandet efter semigrupper av mängder i de ändligdimensionella euklidiska rummenRn, n≥1, vars element inte har Baire-egenskapen, och på studiet av deras egenskaper.
Som bekant är familjen av delmängder avR som har Baire-egenskapen en σ-algebra av mängder som innehåller både de öppna och de magra delmängderna av R. Det finns dock delmängder av R som inte tillhör algebran. Till exempel gäller att varje klassisk Vitali-mängd påR inte har Baire-egenskapen.
Chatyrko har visat att familjen av alla ändliga unioner av Vitali-mängder på reella linjen, och den utvidgning av denna familj som på ett naturligt sätt ges av familjen av magra mängder, är semigrupper av mängder påR, som består av mängder som inte har Baire-egenskapen och som är invarianta under translatio-ner avR.
Genom att använda motsvarigheten till Vitali-mängder när gruppenQ i Vita-lis konstruktion ersätts med en godtycklig uppräknelig tät delgrupp Q avR (vi kallar dessa mängder Vitali-Q-selektorer påR), och Chatyrkos metoder, konstru-erar vi semigrupper av mängder påR relaterade till Q, som består av mängder som inte har Baire-egenskapen och som är invarianta under translationer avR. Vidare studerar vi hur semigrupper relaterade till olika grupper Q förhåller sig med avseende på inklusion, och vi definierar en “supersemigrupp” baserad på alla Vitali-selektorer påR. Denna supersemigrupp består också av mängder som inte har Baire-egenskapen och är invariant under translationer avR. Vi utvidgar och generaliserar sedan dessa resultat från reella linjen till ändligdimensionella euklidiska rumRn, n ≥ 2, och visar på olikheterna i de två fallen n = 1 och n≥2. Dessutom visar vi hur begreppet övertäckningsdimension kan användas för att definiera en mångfald av semigrupper av mängder som inte har Baire-egenskapen.
Acknowledgments
This Licentiate thesis appears in its current form due to the assistance and guid-ance of several people. I would therefore like to offer my sincere thanks to all of them.
First, I would like to express my gratitude to my supervisors Vitalij A. Chatyrko and Mats Aigner for useful comments, remarks and engagement through the learning process until to this Licentiate thesis. They provided encouragement and advise necessary for me to complete this Licentiate thesis and to proceed through the Ph.D program.
In a very special way, I would also like to thank Bengt Ove Turesson, Björn Textorius and all members of the Department of Mathematics for their helps whenever need arises at work. They have directed me through various situ-ations, allowing me to reach this accomplishment. I thank Minani Froduald, Lyambabaje Alexandre and Mahara Isidore for their assistance and guidance and the helpful part they played in my mathematical developments as my teachers.
I would like to thank all my fellow Ph.D students whom I shared so many excellent times at Linköping University. In particular, I would like to thank Anna Orlof for organizing team building activities.
My family has supported and helped me along the learning process by giv-ing encouragement and providgiv-ing the moral and emotional support I needed to complete my thesis. To them, I am eternally grateful.
I wish to acknowledge the financial support I received through Sida/Sarec funded University of Rwanda-Linköping University cooperation. All involved institutions and people are hereby acknowledged.
May the Almighty God richly bless all of you.
Linköping, March 24, 2015 Vénuste NYAGAHAKWA
Contents
1 Introduction 3
1.1 Problem formulation . . . 3
1.2 Summary of main results . . . 4
1.3 Structure of the thesis . . . 6
2 Necessary facts 9 2.1 Algebraic notions in set theory . . . 9
2.2 Some topological concepts . . . 11
2.2.1 Baire Category Theorem . . . 11
2.2.2 Baire property . . . 13
2.2.3 Lebesgue covering dimension . . . 13
3 Algebra of semigroups of sets 15 3.1 Semigroups and ideals of sets . . . 15
3.2 Extension of a semigroup of sets via an ideal of sets . . . 18
4 Semigroups of sets defined by Vitali selectors on the real line 23 4.1 Vitali selectors of the real line . . . 24
4.2 Countable dense subgroups ofR and generated semigroups . . . . 28
4.3 Supersemigroup based on Vitali selectors ofR . . . 31
xii Contents
4.4 Semigroup of non-Lebesgue measurable sets . . . 34 5 Semigroups of sets without the Baire property in finite dimensional
Euclidean spaces 35
5.1 Vitali selectors ofRn . . . 35 5.2 Supersemigroup of Vitali selectors ofRn . . . 39 5.3 Rectangular Vitali selectors ofRn . . . 41 5.3.1 Supersemigroup of rectangular Vitali selectors ofRn . . . . 46 5.4 Semigroups of sets inRn defined by dimension . . . 49
Notation
The principal notation used throughout the text is listed below. N The set of positive integers
Z The set of integers
Q The set of rational numbers R The set of real numbers
Rn The n−dimensional Euclidean space
IntXA Interior of a set A in a topological space X
ClXA Closure of a set A in a topological space X
Yc Complement of a set Y
If Ideal of finite sets
Ic Ideal of countable sets
Icd Ideal of closed and discrete sets
In Ideal of nowhere dense sets
M σ−ideal of meager sets inR
Mn σ−ideal of meager sets inRn
N0 The family of all subsets ofR having Lebesgue measure zero
2 Notation
N The family of all measurable subsets ofR in the Lebesgue sense ∆,∪,∩,\ Standard set operation of symmetric difference, union, intersection
and set difference
dim Lebesgue covering dimension
P (X) The family of all subsets of X
O The family of all open subsets ofR
On The family of all open subsets ofRn
Bp The family of sets with the Baire property on the real line
Bn
p The family of sets with the Baire property inRn
F The family of all countable, dense in the real line subgroups of
(R,+)
Fn The family of all countable, dense in the Euclidean spacesRn, n≥2
subgroups of(Rn,+)
rFn The family of all rectangular subgroups of(Rn,+)
SA Semigroup of sets generated byA
IA Ideal of sets generated byA
V (Q) Family of all Vitali-Q selectors ofR associated to the subgroup Q of
(R,+)
SV (Q) Semigroup of sets generated byV (Q) SVsup Semigroup of sets generated byVsup
Vn(Q) The family of all Vitali-Q selectors ofRn associated to the subgroup
Q of(Rn,+)
SVnQ Semigroup of sets generated byVn(Q)
rVn(Q) The family of all rectangular Vitali-Q selectors ofR associated to the
subgroup Q of(Rn,+)
SrVnQ Semigroup of sets generated by rVn(Q)
SrVn
sup Semigroups of sets generated by rV
n sup
1
Introduction
1.1
Problem formulation
LetR be the set of real numbers andP (R)the family of all subsets ofR. Fur-thermore, let(R, τE) be the real line, i.e. the setR endowed with the topology
τEdefined by all open intervals ofR, andMthe family of all meager subsets of
(R, τE).
An interesting extension ofM, as well as τE inP (R), is the familyBp of all
subsets of(R, τE)possessing the Baire property. Recall ([1]) that a set B∈ Bpif
and only if there are an O∈τE and an M∈ Msuch that B=O∆M.
It is well known (see [1]) that Bp 6= P (R) (for example, each Vitali set V
of R [2] is an element of the complement Bc
p = P (R) \ Bp of Bp in P (R)),
and the familyBpis a σ−algebra of sets, in particular,Bpis closed under finite
unions and finite intersections of sets. Let us also note that the family Bp is
invariant under action of the groupH((R, τE)) of all homeomorphisms of the
real line(R, τE)onto itself, i.e. for each B∈ Bpand each h∈ H((R, τE))we have
h(B) ∈ Bp.
It is easy to see that the familyBc
pis also invariant under action of the group
H((R, τE)) but, unlike the family Bp, Bcp is not closed under finite unions and
finite intersections of sets. It is also well known (cf. [3]) that there are elements of Bc
p with a natural algebraic structure (for example, some subgroups of the
4 1 Introduction
additive group(R,+)of all real numbers). One can pose the following problem ([4]): Do there exist subfamilies ofBc
pwhich are invariant under action of an infinite
sub-group ofH((R, τE))and on which we can define some algebraic structure?
The following simple observation can give an answer to the question. Let G be a non-trivial sufficiently rich subgroupG ofH((R, τE)) and A be
a subfamily ofBc
psuch thatAis invariant under action ofG. If for each n ≥ 2
and each A1, A2,· · ·, An ∈ Awe have A1∪A2∪ · · ·An∈ Bcpthen the familySA
consisting of all finite unions of elements ofAis a semigroup of sets with respect to the binary operation ”∪” union of sets, which is invariant under action ofG
and which is inBc p.
In [5], Chatyrko proved that any union of finitely many Vitali sets is an el-ement ofBc
p. It is easy to see that the family V of all Vitali sets is invariant
under action of the group τ(R) of all translations of R. Hence, by the obser-vation above, the family SV of all finite unions of Vitali sets is a semigroup
of sets with respect to the operation "∪", which is invariant under action of
τ(R)and which is in Bcp. Furthermore, in [4] Chatyrko proved that the family
SV∆M = {U∆M : U∈ SV, M∈ M}is also a semigroup of sets with the respect
to the operation "∪", which is invariant under action of τ(R)and which is inBc p.
The goals of this thesis are the following.
(a) In the realm of P (R) to find families of sets (different from the families SV
and SV∆Mmentioned above) which are semigroups of sets with respect to the
operation ”∪”, which are invariant under action of τ(R)and which are inBc p.
(b) To extend the results of (a) to Euclidean spacesRn, n≥2.
1.2
Summary of main results
Let X be a non-empty set and letP (X)be the family of all subsets of X. For families of setsAandBinP (X), we define two operations:
A ∪ B = {A∪B : A∈ A, B∈ B},
A∆B = {A∆B : A∈ A, B∈ B}.
1.2 Summary of main results 5
Moreover, ifD ⊂ P (X)then bySD we mean the family of all finite unions of
elements ofD.
In Chapter 3, we have obtained the following results (Proposition 3.2 and Proposition 3.4):
(a) IfS is a semigroup of sets with respect to the operation ”∪” andI is an ideal of sets inP (X), then the familiesS ∪ I,S∆I are semigroups of sets with respect to the operation ”∪” andS ⊂ S ∪ I ⊂ S∆I.
(b) LetI be an ideal of sets andA,B ⊂ P (X)such thatA ∩ I = ∅ and for each element U ∈ SA and each non-empty element B∈ Bthere is an element A∈ A
satisfying A⊂B\U. Then(SA∆I ) ∩ (SB∆I ) =∅.
In Chapter 4, the main attention was paid to the case when X=R.
Let Q be a countable, dense in the real line, subgroup of (R,+), V (Q) be the family of all Vitali Q-selectors ofR associated to Q (analogues of Vitali sets, considered in [3], when the set Q of rationals is substituted by Q) and I any subideal of M. Using the Chatyrko’s method, the results from the previous chapter and the observation that τE = SτE, we have proved (Proposition 4.3) that:
(c) The families SV (Q),SV (Q)∪ I and SV (Q)∆I are semigroups of sets with re-spect to the operation ” ∪” andSV (Q) ⊂ SV (Q)∪ I ⊂ SV (Q)∆I. Moreover,
SV (Q),SV (Q)∪ I andSV (Q)∆I are invariant under action of τ(R), and consist of sets without the Baire property.
We have also observed that in the family{SV (Q) : Q ⊂ R}there is no element which contains all others (Proposition 4.8). So we consider the familyVsupof all
Vitali Q-selectors of R, where Q is varied, and the semigroupSVsup which we call a supersemigroup of Vitali selectors. The supersemigroupSVsupcontains the semigroupSV (Q) for each Q.
In the same way as above we have proved (Theorem 4.1) that:
(d) The familiesSVsup,SVsup∪ IandSVsup∆I are semigroups of sets with respect to the operation ”∪” andSVsup ⊂ SVsup∪ I ⊂ SVsup∆I. Moreover,SVsup,SVsup∪
I and SVsup∆I are invariant under action of τ(R), and consist of sets without the Baire property.
6 1 Introduction
In Chapter 5, the main attention was paid to the case when X = Rn, n≥ 2. Let Q be a countable, dense in the Euclidean spaceRn, subgroup of(Rn,+),
Vn(Q)be the family of all Vitali Q-selectors ofRn associated to Q (analogues of
Vitali Q-selectors for the real line, see also [3]) andI any subideal of the family
Mnof all meager sets ofRn. Using the results and technique from the previous
chapters we have proved statements which are similar to (c) and (d).
Moreover, we have pointed out a special case of Vitali Q-selectors of Rn called rectangular Vitali selectors ofRn, n ≥ 2, related to groups Q which are products of n many countable dense in the real line groups. Our rectangular Vitali selectors are supposed to be products of n many Vitali selectors of the real line. We have extended our theory to the special case. Since the rectangular products are somewhat less complicated than general ones, we could obtain more information about them.
A part of the mentioned results can be found in [4] and [6].
1.3
Structure of the thesis
This thesis contains five chapters which are structured as follows:
The first chapter gives a brief description of the problem under investigation and a brief summary of the obtained results.
The second chapter introduces basic concepts, terminology and statements which will be needed for a better understanding of the subsequent chapters.
The third chapter treats the theory of semigroups of sets with respect to the operation of union of sets. Through various examples, we describe the behaviour of semigroups of sets with the respect to several binary operations. Furthermore, we present a way of searching pairs of semigroups of sets without common elements.
The fourth chapter develops the theory of semigroups of sets without the Baire property on the real line. These semigroups are constructed by using Vitali Q-selectors ofR and subideals of the ideal of meager subsets of R. They are invariant under translations of the real line and they consist of sets without the Baire property.
1.3 Structure of the thesis 7
the finite-dimensional Euclidean spacesRn, n ≥ 2. Besides the semigroups of sets generated by ordinary Vitali selectors, the chapter treats also semigroups of sets generated by rectangular Vitali selectors of Rn, n ≥ 2. Rectangular Vitali selectors are a special case of the ordinary ones. In both cases, the generated semigroups of sets are invariant under translations of Rn and they consist of sets without the Baire property. The chapter ends by pointing out the role of dimension in defining different semigroups of sets without the Baire property.
2
Necessary facts
The purpose of this chapter is to recall concepts and terminology which will be used in the subsequent chapters.
2.1
Algebraic notions in set theory
In this section, we shall 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 denoted by capital script letters likeS,Oand so forth.
For a more detailed information, we refer the reader to one of the references [7], [8] or [9].
Let X be a non-empty set and letP (X)be the family of all subsets of X. Definition 2.1. A non-empty familyR ⊂ P (X)of sets is called a ring of sets on X if A∆B∈ Rand A∩B∈ Rwhenever A∈ R, B∈ R.
Since A∪B= (A∆B)∆(A∩B), A\B= A∆(A∩B), we have also A∪B∈ R
and A\B ∈ Rwhenever A∈ R, B∈ R. Thus a ring is a family of sets closed under the operations of taking unions, intersections, differences and symmetric differences. A ring of sets must contain the empty set∅, since A\A=∅.
Definition 2.2. LetA ⊆ P (X)be a ring of sets on X. If X ∈ A, the familyAis called an algebra of sets on X.
10 2 Necessary facts
From this definition, a ring of sets is an algebra if and only if it closed under taking the operation of complement.
Example 2.1
(i) The family of all finite subsets of X is a ring on X but not an algebra on X unless X is finite.
(ii) LetR be the set of real numbers. The family of all bounded subsets of R is a ring onR but not an algebra.
Definition 2.3. (a) A ringR ⊂ P (X)is called a σ-ring of sets on X if it is closed under countable unions, i.e. it contains the union S =S∞
n=1An whenever it
contains the sets A1, A2, . . . .
(b) A σ-ringA ⊂ P (X)is called a σ-algebra of sets on X if X∈ A. From the De Morgan formula T∞
n=1An = X\S∞n=1(X\An), it follows that
each σ-algebra is also closed under countable intersection of sets.
Note that that a σ−algebra can be defined as an algebra closed under count-able unions.
Example 2.2
For a set X, the family of all countable subsets of X is a σ−ring. It will be a
σ−algebra if X is countable.
Definition 2.4. LetA ⊂ P (X). The smallest σ−algebra of sets on X containing
Ais called a σ−algebra generated by the familyA. Example 2.3
LetR be the real line, i.e. the set R endowed with the standard metric ρ defined by ρ(x, y) = |x−y| for all x, y ∈ R and N be the family of all measurable subsets ofR in the Lebesgue sense. The familyN is a σ−algebra of sets which is generated byN0∪ O, whereN0is the family of all subsets ofR having Lebesgue
2.2 Some topological concepts 11
Definition 2.5. (a) A familyI ⊂ P (X)of sets is called an ideal of sets on X, if it satisfies the following two 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 2.4 Let A⊆X. Then
(i) the familyI (A) = {B : B⊆A}is a σ−ideal of sets on X,
(ii) the familyIcof all countable subsets of X forms a σ−ideal of sets on X,
(iii) the familyIf of all finite subsets of X forms an ideal of sets on X, but not a σ−ideal of sets, whenever X is infinite.
2.2
Some topological concepts
2.2.1
Baire Category Theorem
Let X be a topological space and let A be a subset of X.
Recall that a neighborhood of a point x ∈ X is any open subset U of X containing x. The point x∈X is a limit point of A if(U\ {x}) ∩A6=∅ for every neighborhood U of x. The derived set of A, denoted by Ad, is the set of all limit points of A.
Definition 2.6. A subset A of X is said to be closed and discrete if and only if Ad=∅.
Note that each subset of a closed and discrete subset of X (resp. each finite union of closed discrete subsets of X) is also a closed discrete subset of X. Thus the family of all closed and discrete subsets of X forms an ideal of sets, denoted byIcd.
12 2 Necessary facts
Definition 2.7. A subset A⊂X is called a nowhere dense set in X if IntX(ClX(A)) =∅.
It is easy to see that every subset of a nowhere dense set is a nowhere dense set, and the union of finitely many nowhere dense sets is again a nowhere dense set. Thus, the family of nowhere dense sets in a given topological space forms an ideal of sets, denoted byIn.
Example 2.5
Every finite subset of the real lineR, the set Z of all integers and the Cantor set, are nowhere dense subsets ofR.
Definition 2.8. A subset A⊂X is said to be dense in X if ClX(A) =X.
Example 2.6
(1) The setQ of all rational numbers is a dense subset of R.
(2) The setZ(√2) = {n+√2m : n∈Z, m∈Z}is a dense subset ofR [10].
Remark 2.1. Note that a countable union of nowhere dense sets is not necessarily a nowhere dense set. For example, the setQ of rationals is a union of countably many nowhere dense sets inR, but IntR(ClR(Q)) =R.
Definition 2.9. A subset A⊂ X is meager (or of first category) if A is the union of countably many nowhere dense sets. Any set that is not meager is said to be nonmeager (or of second category).
Example 2.7
The setQ of rationals numbers is a meager subset of the real line R.
In a given topological space, the family of all meager sets forms a σ−ideal of sets. The σ−ideal of meager sets will be denoted byMin our further consider-ations.
2.2 Some topological concepts 13
Theorem 2.1(Baire Category Theorem). Let X be a complete metric space. Then X can not be covered by countably many nowhere dense subsets. Moreover, the union of countably many nowhere dense subsets of X has a dense complement.
Recall that the real lineR is a complete metric space. So by Baire Category Theorem,R is of the second category.
Similarly, the Euclidean space Rn, n ≥ 1, i.e. the set Rn endowed with the metric ρ(x, y) =q∑ni=1(xi−yi)2, where x= (x1,· · ·, xn)and y= (y1,· · ·, yn),
is a complete metric space. SoRnis of the second category.
Remark 2.2. On the real lineR the ideals of setsIf,Icd,Ic,InandMsatisfy the
inclusionsIf ( Icd( Ic( MandIf ( Icd( In( M.
Note that Ic and In are not comparable in the sense of inclusion onR. In
fact, the Cantor set is uncountable and nowhere dense subset ofR while the set Q of rationals numbers is a countable dense subset of R.
2.2.2
Baire property
In this subsection, X is assumed to be a topological space.
Definition 2.10. A subset A ⊂ X is said to have the Baire property if it can be represented in the form A = O∆M, where O is an open set of X and M is a
meager set of X. (Recall that O∆M= (O\M) ∪ (M\O))
Note that a subset A⊂X has the Baire property in X if and only if there is an open set O of X and two meager sets M, N of X such that A= (O\M) ∪N.
The family of all sets with the Baire property in a topological space X will be denoted byBpin our further considerations.
Recall that the familyBpis a σ-algebra of sets. In particular, each open set of
X and each meager set of X have the Baire property. Thus, the σ-algebra Bp is
the smallest σ-algebra containing all open and all meager sets in X.
2.2.3
Lebesgue covering dimension
In this subsection, we present some basic properties of Lebesgue covering di-mension dim.
Let X be a topological space and let A = {Aα}α∈Γ be a family of subsets of X, whereΓ is an index set.
14 2 Necessary facts
Definition 2.11. (i) The order of the family A = {Aα}α∈Γ of subsets, not all empty, of X, is the largest integer n for which there exists a subset I of Γ with n+1 elements such that ∩α∈IAα is non-empty, or ∞ if there is no such largest integer.
(ii) The familyA = {Aα}α∈Γis a cover of X if S
α∈ΓAα=X.
(iii) A cover B is a refinement of another cover A of the same space X, if for every B∈ Bthere exists A∈ Asuch that B⊂A.
Definition 2.12. Let X be a topological space. Then
H dim X= −1 if and only if X=∅.
H dim X ≤n if each finite open cover of X has an open refinement of order not exceeding n.
H dim X=n if it is true that dim X≤n but it is not true that dim X≤n−1.
H dim X=∞ if for every integer n it is false that dim X≤n.
If dim X=n, then X is called the n−dimensional topological space.
Recall that a space X is said to be separable if it contains a countable dense subset. It 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 about Lebesgue cov-ering dimension are summarized in the following theorems [13].
Theorem 2.2(Fundamental Theorem of Dimension). For every natural number n, we have dimRn =n.
Theorem 2.3(Monotonicity). If A is a subspace of a separable metrizable space X, then dim A≤dim X.
Theorem 2.4(Countable Sum Theorem). Let X be a separable metrizable space and X=S∞
i=1Fiwhere Fiis closed in X for each i. If dim Fi ≤n for each i, then dim X≤n.
Theorem 2.5 (Product Theorem). Let X and Y be non-empty separable metrizable spaces. Then dim(X×Y) ≤dim X+dim Y.
Theorem 2.6(Brouwer Dimension Theorem). Let X⊆Rn. Then dim X=n if and
3
Algebra of semigroups of sets
In this chapter we introduce the notion of a semigroup of sets. Then we look at the behaviour of semigroups of sets under some binary operations. Additionally, we present a way to extend a given semigroup of sets to another one by the use of ideals of sets. Before ending the chapter, we state and prove a proposition which will be used in searching of pairs of semigroups of sets without common elements. The results of this chapter were taken from the article [6].
Below X is assumed to be a non-empty set and P (X) is the family of all subsets of X.
3.1
Semigroups and ideals of sets
Families of sets, like rings of sets or algebra of sets, are of fundamental impor-tance in Topology and Analysis, and their properties are well known (see [9], [14]). In this section, we will consider another families of sets, namely, semi-groups of sets and ideal of sets, and prove some statements about them.
Definition 3.1. A non-empty set S is called a semigroup if there is a binary operation? : S × S −→ S for which the associativity law is satisfied, i.e. the equality (x?y) ?z = x? (y?z) holds for all x, y, z ∈ S. The semigroup S is called abelian if x?y=y?x for all x, y∈ S.
16 3 Algebra of semigroups of sets
Consider a family of sets S ⊆ P (X) such that for each pair of elements A, B ∈ S we have A∪B∈ S. Since the union of sets is both commutative and associative, such a family of sets will be an abelian semigroup with the respect to the operation of union of sets. This observation leads to the following definition. Definition 3.2. A non-empty family of setsS ⊆ P (X) is called a semigroup of sets on X if it is closed under finite unions.
Remark 3.1. Using the definition of a semigroup of sets, we can redefine the notion of an ideal of sets in the following way: a non-empty familyI ⊆ P (X)is an ideal of sets on X iff it is a semigroup of sets on X and if A∈ I and B⊆ A then B∈ I.
LetA ⊂ P (X). PutSA ={Sni=1Ai : Ai∈ A, n∈N}andIA = {B∈ P (X):
there is A∈ SA such that B⊆A}.
The following proposition is evident.
Proposition 3.1. The familySA is a semigroup of sets on X and the familyIA is an
ideal of sets on X.
We will callSAthe semigroup of sets generated by the familyAandIA will be
called the ideal of sets generated by the familyA.
Let us define three binary operations on subfamilies ofP (X)as follows. IfA,B ⊂ P (X)then
(1) A ∪ B = {A∪B : A∈ A, B∈ B}; (2) A∆B = {A∆B : A∈ A, B∈ B};
(3) A ∗ B = {(A\B1) ∪B2: A∈ A; B1, B2∈ B}
where∪,∆ and\are the usual union, symmetric difference of sets and difference of sets, respectively.
For the defined operations, we observe the following:
(i) Since the union and the symmetric difference of sets are commutative op-erations, we have A ∪ B = B ∪ A and A∆B = B∆A. From the fact that A∪B = (A\B) ∪B = (B\A) ∪A, we have A ∪ B ⊂ A ∗ B and
A ∪ B ⊂ B ∗ A. We note that if A and B are both semigroups of sets or ideals of sets, then the familyA ∪ Bis of the same type.
3.1 Semigroups and ideals of sets 17
(ii) As we will see in the following examples, in general for given semigroups of sets A and B, the families A∆B,A ∗ B,B ∗ A do not need to be semi-groups of sets and none of the inclusions A∆B ⊆ A ∪ B, A∆B ⊇ A ∪ B,
A∆B ⊆ A ∗ B,A∆B ⊇ A ∗ B,A ∗ B ⊆ B ∗ Aneeds 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}andB = {B, X}. Note thatA = SA,B = SBand the families
A ∪ B = {X},A∆B = {∅, A, B, X},A ∗ B = {B, X},B ∗ A = {A, X} are semi-groups of sets. Moreover, none of the following inclusionsA∆B ⊆ A ∪ B,
A∆B ⊆ A ∗ B,A ∗ B ⊆ B ∗ AandB ∗ A ⊆ A ∗ Bholds.
In our further considerations, the notation Ycmeans the complement of a set Y in the set X.
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 haveSA∪ SB =
{∅, A1, A2, B1, B2,{1}c,{2}c,{3}c,{4}c, 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 andSA∪ SB⊆ SA∆SBdo not hold. We note also that none
of the families SA∆SB,SA∗ SB and SB∗ SA are semigroups of sets. In fact,
A1, D ∈ SA∆SB but A1∪D = {4}c ∈ S/ A∆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}andSB= {∅, B1, B2, X}. First we will show that the family
SA∗ SBis not a semigroup of sets. It is enough to prove that the set
C = ((A1\B1) ∪∅) ∪ ((A2\B2) ∪∅) ∈ S/ A∗ SB. It is clear that the set C is
18 3 Algebra of semigroups of sets
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 S2is 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 B1or B2), 0 (if S2 = X) or 9 (if S2 = ∅). We have
again a contradiction that proves the statement.
Further note that SB∗ SA = {A1, A2,{1}c,{3}c,{7}c,{9}c, X} = SA∪ SB.
Hence, the familySB∗ SAis a semigroup of sets.
Remark 3.2. The binary operation∗is not commutative, neither is associative, in general. This can be observed from the following example.
Example 3.4
Let A and B be two non-empty subsets of X such that A ( B. Consider the
families of setsA = {∅, A},B = {B}and C = {∅, X}. Then we haveA ∗ B = {B}whileB ∗ A = {B, B\A}. It is clear also that(A ∗ B) ∗ C = {∅, B, X}while
A ∗ (B ∗ C) = {∅, A, B, X}.
3.2
Extension of a semigroup of sets via an ideal of
sets
In this section, we present a way of obtaining a new semigroup of sets from an old one by the use of an ideal of sets.
Proposition 3.2. LetS be a semigroup of sets andI be an ideal of sets. Then the family
S ∗ I is a semigroup of sets.
Proof. Take two elements U1 and U2 of S ∗ I. Then U1 = (S1\I
0 1) ∪I 00 1 and U2 = (S2\I 0 2) ∪I 00
2) for some sets Si ∈ S and I
0
i, I
00
i ∈ I where i = 1, 2. We
need to show that the set U =U1∪U2 ∈ S ∗ I. In fact U = ((S1\I
0 1) ∪I 00 1) ∪ ((S2\I 0 2) ∪I 00 2) = (S1\I 0 1) ∪ (S2\I 0 2) ∪ (I 00 1∪I 00
2). SinceI is an ideal of sets, then
the set I2 = I 00 1 ∪I 00 2 ∈ I. It follows that U = ((S1∩I 0c 1) ∪ (S2∩I 0c 2))cc∪I2 =
3.2 Extension of a semigroup of sets via an ideal of sets 19 ((S1∩I 0c 1)c∩ (S2∩I 0c 2)c)c∪I2= ((Sc1∪I 0 1) ∩ (Sc2∪I 0 2))c∪I2= ((S1c∩Sc2) ∪ (S1c∩ I20) ∪ (Sc2∩I10) ∪ (I10∩I20))c∪I2. Put I1= (S1c∩I 0 2) ∪ (Sc2∩I 0 1) ∪ (I 0 1∩I 0 2)and note that U= ((Sc1∩Sc2)c∩I1c) ∪I2= ((S1∪S2) ∩I1c) ∪I2= ((S1∪S2) \I1) ∪I2. It is
easy to see that S1∪S2∈ S and I1, I2∈ I. Hence, U∈ S ∗ I.
The following proposition shows the relationship in the sense of inclusion between the defined operations when they are applied on semigroups of sets and ideals of sets.
Proposition 3.3. LetS be a semigroup of sets andI 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 ∗ Sand S=S∪∅∈ S ∪ I. Thus,S ∗ I ⊃ S∆I ⊃ S ∪ I ⊃ SandI ∗ 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 andI ∗ 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). HenceI ∗ (I ∗ S ) ⊂ I ∗ S. The opposite
inclusion is evident.
Corollary 3.1. LetS be a semigroup of sets andI be an ideal of sets. Then (a) The familiesS∆I,I ∗ S are semigroups of sets;
(b) (I ∗ S ) ∗ I = I ∗ (S ∗ I ) = S ∗ I
Proof. As S∆I = S ∗ I, by Proposition 3.2, the family S∆I is a semigroup of sets. From the observation (i) made above and the equalityS ∪ I = I ∗ S, the family I ∗ S is also a semigroup of sets. This proves item (a). The item (b) follows by observing that S ∗ I = (S ∗ I ) ∗ I ⊃ (I ∗ S ) ∗ I ⊃ S ∗ I and
20 3 Algebra of semigroups of sets
The following statement is evident.
Corollary 3.2. LetI1,I2be ideals of sets. Then the familyI1∗ I2is an ideal of
sets. Moreover,I1∗ I2= I2∗ I1= I1∆I2= I1∪ I2.
Example 3.5
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}andIB∗ 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, where S is a semigroup of sets andI is an ideal of sets.
For two subfamiliesAandBofP (X), putA ∩ B = {Y : Y∈ Aand Y∈ B}. The next statement is useful in the search of pairs of semigroups of sets without common elements.
Proposition 3.4. LetI be an ideal of sets andA,B ⊂ P (X)such that: (a) A ∩ I =∅ (i.e.AandIhave no common element);
(b) For each element U ∈ SA and each non-empty element B∈ B there is an element
A∈ Asuch 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∩B6=∅;
(2) For each elements I1, I2∈ I, each element U∈ SAand each non-empty B∈ Bwe
have(U∪I1)c∩ (B\I2) 6=∅;
(3) For each elements I1, I2, I3, I4∈ I, each element U∈ SAand each element V∈ SB
we have(U\I1) ∪I26= (V\I3) ∪I4, i.e. (SA∗ I ) ∩ (SB∗ I ) =∅.
Proof. (1) Assume that U∪I ⊃ B for some non-empty element B ∈ B. By (b) there is A∈ Asuch that A⊂B\U. Note that A⊂ (U∪I) \U⊂ I. But this is a contradiction with (a).
(2) Assume that U∪I1 ⊃ B\I2for some non-empty element B ∈ B and some
element I2∈ I. Note that U∪ (I1∪I2) = (U∪I1) ∪I2 ⊃ (B\I2) ∪I2 ⊃ B.
3.2 Extension of a semigroup of sets via an ideal of sets 21
(3) Assume that(U\I1) ∪I2= (V\I3) ∪I4for some elements U∈ SA, V ∈ SB
and I3, I4 ∈ I. If V = ∅, then(U\I1) ∪I2 = I4 and so U ⊂ I1∪I4. But
this is a contradiction with with (a). Hence V 6= ∅. Note that there is a
non-empty element B ∈ B such that B ⊂V. Further observe that U∪I2 ⊃
4
Semigroups of sets defined by Vitali
selectors on the real line
In this chapter, we present diverse semigroups of sets on the real line whose el-ements do not possess the Baire property. These semigroups will be constructed by the use of a concept of Vitali selectors rigorously defined in the coming sec-tion. The Vitali selectors are closely related to the classical Vitali sets on the real line (one should to replace the rationalsQ by any countable dense subgroup Q of the realsR in the Vitali construction). They were considered in [3] and were calledΓ-selectors of R there. It is known (cf. [3]) that each Vitali set (even each Vitali selector) of R does not possess the Baire property. This result was ex-tended by V.A. Chatyrko [5] who showed that finite unions of Vitali sets on the real line also do not possess the Baire property. Furthermore, he observed that the familySV (Q)of all finite unions of Vitali sets, as well as the familySV (Q)∗ M, whereMis the family of all meager sets on the real line, are semigroups of sets, invariant under translations ofR, and the elements of SV (Q)∗ M also do not possess the Baire property.
So in this chapter, we generalize Chatyrko’s results via replacing the Vitali sets by Vitali Q-selectors, where Q is a countable dense subgroup Q ofR, and the familyMby any ideal of sets onR in the formulas SV (Q) andSV (Q)∗ M. We study the relationship between the semigroups for different Q in the sense of inclusion. We observe that in the family of all semigroups of setsSV (Q), where Q is varied, there is no element which contains all others. Furthermore we
24 4 Semigroups of sets defined by Vitali selectors on the real line
consider the familyVsup of all Vitali selectors ofR and the familyS
Vsup which we call the supersemigroup of sets based on Vitali selectors. We will show that the supersemigroupSVsupalso consists of sets without the Baire property and is invariant under translations ofR. Let us note that the semigroupSVsup contains all semigroupsSV (Q).
The results of this chapter were mostly taken from the articles [4] and [6].
4.1
Vitali selectors of the real line
Let R be the real line and Q be a countable, dense in the real line subgroup of (R,+). For an element x ∈ R, denote by Tx the translation ofR by x, i.e.
Tx(y) = y+x for each element y ∈ R. If A is a subset of R and x ∈ R, we
denote Tx(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 equivalence classes. Observe that|I| =c, where c is the cardinality of the continuum, and for each index α∈I and each x ∈Eα(Q)we have Eα(Q) =Qx. So, each equivalence class Eα(Q)is dense inR.
Definition 4.1. A Vitali Q−selector (shortly, a Vitali selector) ofR is any subset V ofR such that|V∩Eα(Q)| =1 for each α∈I.
Note that a Vitali Q−selector is a Vitali set [2], if Q is the set Q of rational numbers.
We continue with simple facts about Vitali selectors. Proposition 4.1. Let V be a Vitali Q−selector ofR.
(i) If q1, q2∈Q and q16=q2then Vq1∩Vq2 =∅. (ii) R=S
q∈QVq.
(iii) The set V is not meager inR.
Proof. (i) Let x ∈ Vq1 ∩Vq2 with q1, q2 ∈ Q and q1 6= q2. Then x can be represented in two ways: x = y+q1 = z+q2 for some y, z ∈ V. But
y−z=q2−q1∈Q implies that y and z are in the same equivalence class.
Since|V∩Eα(Q)| =1 for all α ∈ I, then y =z. This implies that q1= q2.
4.1 Vitali selectors of the real line 25
(ii) If x∈R, then x belongs to a unique equivalence class Eα(Q). Let vαbe the representatative of Eα(Q)in V, i.e. V∩Eα(Q) = {vα}. So, x−vα =q for some q∈Q. It follows that x=vα+q∈Vq.
(iii) If V is a meager in R, then each Vq, q ∈ Q, is a meager subset of R, as
a translation is a homeomorphism. This implies that the real line R is covered by countably many meager sets, and hence it is meager. This is in a contradiction with the Baire Category Theorem.
Lemma 4.1. For each Vitali Q−selector V ofR and each element x ∈R, the set Vxis
also a Vitali Q−selector ofR.
Proof. Let V be an arbitrary Vitali Q−selector and let x∈R.
Since for any different elements v1, v2 of V, we have (v1+x) − (v2+x) =
v1−v2∈R\Q, the elements v1+x and v2+x belong to different equivalence
classes Eα, α∈ I.
Consider a fixed α ∈ I and let vα be the element in V satisfying {vα} = Eα(Q) ∩V. So vα−x ∈ Eβ(Q)for some β ∈ I. Note that there is one element vβ in V such that{vβ} = V∩Eβ(Q). Since vα−x and vβ belong to the same equivalence class Eβ(Q), there is a q∈Q such that vα−x−vβ =q. So vβ+x= vα−q∈Eα(Q). As vβ+x∈Vx, we have|Vx∩Eα(Q)| =1 for all α∈I.
The family of all Vitali Q−selectors ofR associated to the subgroup Q will be denoted byV (Q)and SV (Q) will denote the semigroup of sets generated by
V (Q)(Chapter 3, Section 3.1).
Proposition 4.2. The familiesV (Q)andSV (Q)are invariant under translations ofR. Proof. Let V ∈ V (Q). By Lemma 4.1, it follows that for each element x∈R, we
have Vx∈ V (Q). So, the familyV (Q)is invariant under translations ofR. Since
the familySV (Q) consists of all finite unions of elements ofV (Q), then it is also invariant under translations ofR.
Lemma 4.2. For each U ∈ SV (Q) and each non-empty open set O of R, there is an element V ∈ V (Q)such that V⊂O\U.
Proof. Let U ∈ SV (Q) and O be a non-empty open set of R. So U = Sn
i=1Vi
where Vi ∈ V (Q). To continue with the proof, first we show the following useful
26 4 Semigroups of sets defined by Vitali selectors on the real line
Claim 4.1.1. For each element α∈I, we have Eα(Q) ∩ (O\U) 6=∅.
Proof. From the density of each equivalence class Eα(Q)in the real line, we have
|Eα(Q) ∩O| = ℵ0for all α∈ I. By the definition of a Vitali Q−selector, we have
|Eα(Q) ∩U| = |Eα(Q) ∩ ( Sn
i=1Vi)| = |Sni=1(Eα(Q) ∩Vi)| ≤n for all α∈ I. These two facts show that Eα(Q) ∩ (O\U) 6=∅.
For each equivalence class Eα(Q), α ∈ I, choose one element yα in the set Eα(Q) ∩ (O\U). The set V of such elements yα is a Vitali Q−selector of R. Moreover, V⊂O and V∩U=∅. Hence V⊂O\U.
LetO be the family of all open subsets ofR. Note thatOis a semigroup of sets andSO = O.
Proposition 4.3. LetI be an ideal of subsets ofR. Then the following statements hold. (i) The families SV (Q),I ∗ SV (Q) and SV (Q)∗ I are semigroups of sets such that
SV (Q)⊂ I ∗ SV (Q) ⊂ SV (Q)∗ I.
(ii) IfV (Q) ∩ I =∅, then(SV (Q)∗ I ) ∩ (O ∗ I ) =∅. In particular,SV (Q)∩ (O ∗ I ) =∅.
(iii) IfIis invariant under translations ofR, then the familiesI ∗ SV (Q)andSV (Q)∗ I
are also invariant under translations ofR.
Proof. (i) The family SV (Q) is a semigroup of sets by Proposition 3.1. By Proposition 3.2 and Corollary 3.1, the familiesSV (Q)∗ I andI ∗ SV (Q) are also semigroups of sets. The inclusion follows from Proposition 3.3. (ii) To prove the equality(SV (Q)∗ I ) ∩ (O ∗ I ) =∅, we apply Proposition 3.4,
together with Lemma 4.2. Namely, the familiesAandBin Proposition 3.4 are considered as the families V (Q) and O, respectively, and Lemma 4.2 plays the same role as condition (b) in Proposition 3.4.
The particular case follows from the inclusionSV (Q)⊂ SV (Q)∗ I.
(iii) The invariance of the familiesI ∗ SV (Q) and SV (Q)∗ I under translations ofR follows from Proposition 4.2 and the assumption made onI.
4.1 Vitali selectors of the real line 27
Let M be the σ−ideal of meager sets in R and let Bp be the family of all
subsets ofR having the Baire property.
Observe thatV (Q) ∩ M =∅ (Proposition 4.1 (iii)),Bp= O ∗ M(see
Defini-tion 2.10) andMis invariant under translations ofR.
Corollary 4.1. The familiesSV (Q),M ∗ SV (Q) andSV (Q)∗ Mare semigroups of sets such thatSV (Q) ⊂ M ∗ SV (Q) ⊂ SV (Q)∗ M. They are invariant under translations ofR, and consist of sets without the Baire property. In particular, the family SV (Q)
consists of sets without the Baire property.
Proof. Note only that by Proposition 4.3 (ii), we have the equality
(SV (Q)∗ M) ∩ (O ∗ M) =∅. SinceBp= O ∗ M, thenSV (Q)∗ M ⊂ P (R) \ Bp.
In particular, from the inclusionSV (Q)⊂ SV (Q)∗ M, it follows that
SV (Q)⊂ P (R) \ Bp.
LetIf (resp. Ic,Icd orIn) be the ideal of finite (resp. countable, closed and
discrete or nowhere dense) subsets ofR. Recall that for these ideals of sets, we have the inclusions If ( Icd ( Ic ( Mand Icd ( In ( M. Note that these
ideals are invariant under translations ofR.
Corollary 4.2. LetI beIf,Ic,Icdor In. Then the familiesI ∗ SV (Q)andSV (Q)∗ I
are semigroups of sets such thatSV (Q) ⊂ I ∗ SV (Q) ⊂ SV (Q)∗ I. They are invariant under translations ofR and consist of sets without the Baire property.
Proof. The results follows from Proposition 4.3 and the mentioned above inclu-sions.
Remark 4.1. From the inclusions If ( Icd ( Ic ( M and Icd ( In ( M, it
follows also that:
(a) SV (Q)∗ If ⊂ SV (Q)∗ Icd⊂ SV (Q)∗ Ic⊂ SV (Q)∗ Mand
SV (Q)∗ Icd⊂ SV (Q)∗ In⊂ SV (Q)∗ M.
(b) If ∗ SV (Q)⊂ Icd∗ SV (Q)⊂ Ic∗ SV (Q)⊂ M ∗ SV (Q)and
Icd∗ SV (Q)⊂ In∗ SV (Q)⊂ M ∗ SV (Q).
28 4 Semigroups of sets defined by Vitali selectors on the real line
Example 4.1
Let Q=Q. Observe that for each element A∈ SV (Q)∗ If, we have|A∩Q| <∞. In fact, if A∈ SV (Q)∗ If then A= (U\I1) ∪I2where U∈ SV (Q)and
I1, I2 ∈ If. Recall that U = Sni=1Vi, where Vi ∈ V (Q). Thus, |A∩Q| = |((U\
I1) ∪I2) ∩Q| ≤ |(U∪I2) ∩Q| = |(U∩Q) ∪ (I2∩Q)| ≤ |U∩Q| + |I2∩Q| ≤
n+ |I2| <∞.
This implies that the semigroups of sets SV (Q)∗ If and SV (Q)∗ Icd are not equal. In fact, let V∈ V (Q)and letZ be the set of all integers. Since Z is a closed and discrete subset ofR, we have V∪Z∈ SV (Q)∗ Icd. But|(V∪Z) ∩Q| = ℵ0.
So, by the above observation, V∪Z /∈ SV (Q)∗ If.
Proposition 4.4. LetIbe an ideal of sets such thatV (Q) ∩ I =∅. Then each element
of the familySV (Q)∗ I is zero-dimensional. In particular, each element of the family
SV (Q) is zero-dimensional.
Proof. Let A ∈ SV (Q)∗ I. Then A = (U\F) ∪E for some U ∈ SV (Q) and E, F ∈ I. The equality V (Q) ∩ I = ∅ implies that A6= ∅. Since ∅ 6= A ⊂ R
then 0≤dim A≤1, by the monotonicity property of dimension.
Assume that dim A = 1. By the Brouwer-Dimension Theorem, there must exist a non-empty open set O inR such that O ⊆ A. But A = (U\F) ∪E ⊆
U∪E. By Lemma 4.2, there exists V ∈ V (Q) such that V ⊂ O\U. So, V ⊂
O\U⊆ (U∪E) \U⊆ E which implies that V∈ I. This is a contradiction. So dim A=0.
The particular case follows from the inclusionSV (Q)⊂ SV (Q)∗ I. Corollary 4.3. Each element of the familySV (Q)∗ Mis zero-dimensional. Proof. The statement follows from Proposition 4.4.
4.2
Countable dense subgroups of
R and generated
semigroups
Let Q1and Q2be subgroups of(R,+). Define Q1+Q2= {q1+q2: qi∈ Qi, i=
1, 2}. We observe that the sum Q1+Q2is a subgroup of(R,+)and both Q1and
4.2 Countable dense subgroups ofR and generated semigroups 29
(i) If each Qi, i =1, 2 is countable then the subgroup Q1+Q2is countable;
(ii) If one of the subgroups Qi, i=1, 2 is dense then so is Q1+Q2.
Let F be the family of all countable, dense in the real line subgroups of the additive group(R,+).
Proposition 4.5. For each Q1∈ F, there is a Q2∈ F such that Q1(Q2.
Proof. Let Q1 ∈ F. Since Q1 is a countable subset of R and the set R is
un-countable, we have R\Q1 6= ∅. Consider an element x ∈ R\Q1 and set
Q2 = Q1+xZ = {q+nx : q ∈ Q1, n∈ Z}, whereZ is the additive group of
all integers. It is clear that Q2is a countable subgroup of (R,+)and Q1 ⊂Q2.
The subgroup Q2 is dense onR (it contains a dense subset Q1ofR) and hence
Q2∈ F. Moreover, Q1(Q2, since x∈Q2\Q1.
Proposition 4.6. Let Q1, Q2be elements ofF such that Q1(Q2.
ThenSV (Q
1)∩ V (Q2) =∅. In particular,V (Q1) ∩ V (Q2) =∅. Proof. Assume that there exists V ∈ SV (Q
1)∩ V (Q2). Let Eα(Q2)be an equiva-lence class with the respect to the subgroup Q2. So,
|V∩Eα(Q2)| =1 (4.1)
Since Q1 ( Q2, we have |Q2/Q1| > 1, where Q2/Q1 is the factor group of Q2
by Q1. Note that Eα(Q2) = Sβ∈AαEβ(Q1), where Eβ(Q1) are distinct equiva-lence classes with the respect to the subgroup Q1 and |Aα| = |Q2/Q1| > 1. It
follows that |V∩Eα(Q2)| = |V∩ ( S
β∈AαEβ(Q1))| = | S
β∈Aα(V∩Eβ(Q1))| = ∑β∈Aα|V∩Eβ| = |Aα| >1. This is in contradiction with the Equality 4.1. So, we must haveSV (Q1)∩ V (Q2) =∅.
The particular case follows the inclusionV (Q1) ⊂ SV (Q1).
Remark 4.2. Let Q1, Q2be elements ofF such that Q1( Q2and Q2/Q1be the
factor group of Q2by Q1. We have either 1< |Q2/Q1| <∞ or|Q2/Q1| = ℵ0.
Proposition 4.7. For each Q1 ∈ F there is a Q2 ∈ F such that Q1 ( Q2 and
|Q2/Q1| = ℵ0.
Proof. Let Q1 ∈ F. We can apply repeatedly the Proposition 4.5 to obtain a
30 4 Semigroups of sets defined by Vitali selectors on the real line
Set Q1 = Q1. By Proposition 4.5 we can get Q2 ∈ F such Q1 ( Q2, where
Q2=Q1+x
1Z, x1∈R\Q1. In the same way we can get Q3∈ F such Q2(Q3
where Q3=Q2+x2Z, x2∈R\Q2, and so on.
By this procedure, we get a sequence{Qk}∞k=1of elements inF with
Q1
(Q2(Q3( · · · (Qk−1 (Qk⊆ · · ·, where Qk at the kth step is given by
Qk = Qk−1+xk−1Z and xk−1 ∈ R\Qk−1. The inequality Qk 6= Qk+1 is clear
since xk ∈Qk+1\Qk.
Put Q2 = S∞k=1Qk. It is evident that Q2 is a subgroup of(R,+). Besides
that, Q2is countable (it is a countable union of countable sets) and dense onR
(it contains dense subsets Qk, k=1, 2,· · · ofR). Hence Q2∈ F.
To prove that |Q2/Q1| = ℵ0, we will observe that for each pair of distinct
elements xnand xmin the sequence{xk}∞k=1, we have(Q1+xn) ∩ (Q1+xm) =∅.
For, let y∈ (Q1+xn) ∩ (Q1+xm)for some n>m. Then y=q1+xn=q2+
xmfor some q1, q2∈Q1. So xn = (q2−q1) +xm∈ Q1+xm⊂Q1+xmZ⊂Qm+1.
As n>m, we must have xn∈Qm+1⊆Qn. But, by the construction xn ∈R\Qn.
We have a contradiction. So(Q1+xn) ∩ (Q1+xm) =∅.
This observation implies that Q1+x1, Q1+x2, . . . , Q1+xk, . . . are different
elements of Q2/Q1. Hence,|Q2/Q1| = ℵ0.
Proposition 4.8. Let Q1, Q2be elements ofF such that Q1(Q2and|Q2/Q1| = ℵ0.
ThenSV (Q1)∩ SV (Q2)=∅.
Proof. Assume that there exists U ∈ SV (Q1)∩ SV (Q2). Since U∈ SV (Q2), then U can be represented as U=Sn
i=1Viwhere Vi∈ V (Q2)for all i.
Let Eα(Q2)be an equivalence class with the respect to the subgroup Q2. So,
|U∩Eα(Q2)| ≤n. (4.2)
Since|Q2/Q1| = ℵ0, we have Eα(Q2) = Sβ∈AαEβ(Q1), where Eβ(Q1)are dis-tinct equivalence classes with the respect to the subgroup Q1and|Aα| = ℵ0.
Since U ∈ SV (Q
1) then U can be also represented as U =
Sm
k=1Wk where
Wk∈ V (Q1)for all k.
Let W be an arbitrary Vitali Q1-selector among Wi’s making the union U. So
|U∩Eα(Q2)| ≥ |W∩Eα(Q2)| = |W∩ (Sβ∈AαEβ(Q1))| = | S
β∈Aα(W∩Eβ(Q1))| = ∑β∈Aα|W∩Eβ(Q1)| = |Aα| = ℵ0.
4.3 Supersemigroup based on Vitali selectors ofR 31
Corollary 4.4. For each Q1 ∈ F there is a Q2 ∈ F such thatSV (Q2)∩ SV (Q1) =∅. In particular, the family{SV (Q) : Q ∈ F }of all semigroups of Vitali selectors has no element which contains all others.
Proof. Let Q1∈ F. By Proposition 4.7, one can find Q2∈ F such that Q1(Q2
and |Q2/Q1| = ℵ0. Then, Proposition 4.8 implies that SV (Q2)∩ SV (Q1) = ∅. So, there is no Q∗ ∈ F such that the generated semigroup SV (Q∗)contains the
semigroupSV (Q) for each Q∈ F.
4.3
Supersemigroup based on Vitali selectors of
R
In the previous section, we have observed that there is no Q ∈ F such that the corresponding generated semigroupSV (Q) contains all others.
Definition 4.2. PutVsup = {V : V ∈ V (Q), Q ∈ F }. The familyS
Vsup is called the supersemigroup of sets based on Vitali selectors ofR.
For the familySVsup, we have the following observations:
(i) As each element of Vsup is invariant under translation of R (Proposition
4.2), the familySVsupis invariant under translations ofR.
(ii) For each Q ∈ F and each ideal of sets I on R the inclusions V (Q) ⊂ Vsup,S
V (Q) ⊂ SVsup,I ∗ SV (Q) ⊂ I ∗ SVsup and SV (Q)∗ I ⊂ SVsup∗ I evi-dently hold.
(iii) By Propositions 3.2 and Corollary 3.1, the families SVsup∗ I and I ∗ SVsup are semigroups of sets, for each ideal of setsI onR.
Lemma 4.3. For each set U∈ SVsupand each non-empty open set O ofR, there is a set V∈ Vsupsuch that V⊂O\U.
Proof. Let U = Sn
i=1Vi, where Vi ∈ V (Qi)and Qi ∈ F, i = 1,· · ·, n. Note that
the statement is valid when Q1= · · · =Qn (Lemma 4.2). Now we will consider
the general case. Put Q=∑ni=1Qi ={∑i=1n qi : qi ∈Qi}and note that Q∈ F.
Claim 4.3.1. For each x∈R we have|Qx∩ (O\U)| ≥1.
32 4 Semigroups of sets defined by Vitali selectors on the real line
Proof. For n = 1 the statement evidently holds by Lemma 4.2. 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.
Let now Q∗i = {qij: j≥1}, i=1≤n, and qji=qi(j), i≤n; j≤n+1. Observe
that for each i≤ n and each j1,· · ·bji,· · ·, jn (the notation ba means that a is not there) the set{x+qj1
1 + · · · +qik+ · · · +q jn
n : k≥1}consists of countably many
different points (a coset of Qi∗) and only one of them belongs to Vi.
Consider now 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)of B the sum x+qj11+ · · · +qnjn.
Fix i≤n and observe that each interval I(j1,· · ·bji,· · ·, jn) =
{(j1,· · ·, k,· · ·, jn) : k ≤ n+1}of cells contains at most one element of Vi. So
the whole box B contains at most(n+1)n−1elements of Vi.
Summarizing we have at most n(n+1)n−1elements of U in the box B. Since
(n+1)n>n(n+1)n−1for 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 ofR such that V⊂O\U.
The following Theorem summarizes our main results about the supersemi-group of sets based on Vitali selectors on the real line.
Theorem 4.1. LetI be an ideal of subsets ofR. Then the following statements hold. (i) The familiesSVsup,I ∗ SVsupandSVsup∗ Iare semigroups of sets such that
SVsup⊂ I ∗ SVsup ⊂ SVsup∗ I. (ii) IfVsup∩ I =∅ then
(a) (SVsup∗ I ) ∩ (O ∗ I ) =∅. In particular,SVsup∩ (O ∗ I ) =∅.
(b) For each A∈ SVsup∗ I, we have dim A=0.
4.3 Supersemigroup based on Vitali selectors ofR 33
(iii) IfI is invariant under translations ofR, then the familiesI ∗ SVsupandSVsup∗ I are also invariant under translations ofR.
(iv) For each Q∈ F we haveSV (Q)∗ I ⊂ SVsup∗ I.
Proof. (i) The families SVsup,SVsup∗ I and I ∗ SVsup are semigroups of sets by Propositions 3.1, Proposition 3.2 and Corollary 3.1, respectively. The inclusions follow from Proposition 3.3.
(ii) (a) To prove the equality(SVsup∗ I ) ∩ (O ∗ I ) = ∅, we apply Proposition 3.4 and Lemma 4.3: the familiesAandBin Proposition 3.4 are considered as the families Vsup and O, respectively, and Lemma 4.3 plays the same
role as condition (b) in Proposition 3.4. The particular case follows from the inclusionSVsup⊂ SVsup∗ I.
(b) Let A ∈ SVsup∗ I. Then A = (U\M) ∪ N where U ∈ SVsup and M, N ∈ I. The equality Vsup∩ I = ∅ implies that A 6= ∅. Since A ⊂ R,
we must have 0≤dim A≤1.
Assume that dim A =1. By the Brouwer-Dimension Theorem, there must exist a non-empty open set O inR such that O ⊆ A. But A = (U\M) ∪
N⊆U∪N. By Lemma 4.3, there exists V∈ Vsup such that V⊂O\U. So,
V ⊂ O\U ⊆ (U∪N) \U ⊆ N which implies that V is an element of I. This is a contradiction. So dim A=0. The particular case easily follows. (iii) The invariance of the families under translations ofR follows from
Proposition 4.2 and the assumption that I is invariant under translations ofR.
(iv) The result was pointed out in the second observation at the beginning of this section.
Corollary 4.5. The familiesSVsup,M ∗ SVsup and SVsup∗ M are semigroups of sets such thatSVsup ⊂ M ∗ SVsup ⊂ SVsup∗ M. They are invariant under translations of R, and consist of sets without the Baire property. Moreover, for each A∈ SVsup∗ M, we have dim A=0.
Proof. Let us observe that the equality V (Q) ∩ M =∅ for each Q ∈ F implies thatVsup∩ M =∅. Keeping in mind thatB
34 4 Semigroups of sets defined by Vitali selectors on the real line
(SVsup∗ M) ∩ (O ∗ M) =∅, it follows thatSVsup∗ M ⊂ P (R) \ Bp. The other statements of the corollary follow from Theorem 4.1.
Remark 4.3. In the previous Corollary we can substituteM by any ideal from the following listIf,Ic,IcdorIn.
4.4
Semigroup of non-Lebesgue measurable sets
In this section we consider Vitali sets on the real line. Using the operation∗, we produce a semigroup consisting of sets which are not measurable in the Lebesgue sense.
In [15] Kharazishvili proved that each element U of the familySV (Q)is
non-measurable in the Lebesgue sense. LetN be the family of all measurable sets in the Lebesgue sense on the real lineR and letN0 be the family of all sets of
the Lebesgue measure zero. Recall that the familyN0is an ideal of sets (in fact,
a σ−ideal). It follows from Proposition 3.2 and Proposition 3.3 that the families
SV (Q),N0∗ SV (Q)andSV (Q)∗ N0are three different semigroups of sets invariant under translation ofR andSV (Q)⊂ N0∗ SV (Q)⊂ SV (Q)∗ N0.
We have the following generalization of Kharazishvili’s result.
Proposition 4.9. Each element of the family SV (Q)∗ N0 is nonmeasurable in the Lebesgue sense.
Proof. In fact, let A ∈ SV (Q)∗ N0 and assume that A∈ N. By Proposition 3.3
there are an U ∈ SV (Q) and N ∈ N0 such that A = U∆N. It is known that if
A1, A2are sets such that A1∈ N and the set A1∆A2is of the Lebesgue measure
zero, then the set A2must belong to the familyN [16]. But A∆U= (U∆N)∆U=
N, hence U ∈ N. This is a contradiction of Kharazishvili’s result. So we must have A /∈ N.
Question 4.4.1. Is each element U of the family SVsup∗ N0 nonmeasurable in the Lebesgue sense?
5
Semigroups of sets without the Baire
property in finite dimensional
Euclidean spaces
In the previous chapter, we were concerned with semigroups of sets without the Baire property related to Vitali selectors ofR. In this chapter, we consider Vitali selectors ofRn, n≥2. Unlike to the real line, for the Euclidean spacesRn, n≥2, besides ordinary Vitali selectors, there exist also special Vitali selectors called here rectangular Vitali selectors.
Analogously to Chapter 4, we present semigroups of sets without the Baire property generated by these two types of Vitali selectors ofRn, n≥2, etc. Fur-thermore, we explore the role of dimension in defining semigroups of sets on Rn without the Baire property.
The results of this chapter were mostly taken from the articles [4] and [6].
5.1
Vitali selectors of
R
nLet n≥2. Consider(Rn,+)the additive group of real n-tupples and let Q be a countable dense subgroup of(Rn,+), whereRn is endowed with the Euclidean topology.
For x ∈ Rn, denote by T
x the translation of Rn by x, i.e. Tx(y) = y+x for
each y∈ Rn. If A is a subset ofRn and x ∈ Rn, we denote T
x(A)by Ax. It is
