����������� ������ 58 (2006), 125–129

UDK 515.122

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SPACES RELATED TO γ-SETS

Filippo Cammaroto^{1} and Ljubiša D.R. Kočinac^{2}

Abstract. We characterize Ramsey theoretically two classes of spaces which are related to γ-sets.

1. Introduction

The notation and terminology are mainly as in [2]. X will denote an infinite Hausdorff topological space.

Let A and B be sets whose members are families of subsets of an infinite set X. Then (see [7], [4]):

S_{1}(A, B) denotes the selection principle:

For each sequence (A_{n} : n ∈ N) of elements of A there is a sequence (b_{n} : n ∈ N) such that for each n ∈ N, b_{n} ∈ A_{n} and {b_{n} : n ∈ N} ∈ B.

S_{fin}(A, B) denotes the selection hypothesis:

For each sequence (A_{n} : n ∈ N) of elements of A there is a sequence (B_{n} : n ∈ N) of finite (not necessarily non-empty) sets such that for each n ∈ N, B_{n} ⊂ A_{n} and ⋃_{n∈N} B_{n} is an element of B.

The symbol G_{1}(A, B) [7] denotes an infinitely long game for two players, ONE and TWO, which play a round for each positive integer. In the n-th round ONE chooses a set A_{n} ∈ A, and TWO responds by choosing an element b_{n} ∈ A_{n}. TWO wins a play (A_{1}, b_{1}; ⋯; A_{n}, b_{n}; ⋯) if {b_{n} : n ∈ N} ∈ B; otherwise, ONE wins.

If ONE does not have a winning strategy in the game G_{1}(A, B), then the selection hypothesis S_{1}(A, B) is true, but the converse need not be always true. In many cases the game characterizes the corresponding selection principle.

AMS Subject Classification: 54D20, 05C55, 03E02, 91A44.

Keywords and phrases: Selection principles, Ramsey theory, game theory, ω-cover, k-cover, γ-cover, γ_{k}-cover, γ-set, k-γ-set, γ′_{k}-set.

1Supported by MURST-PRA 2000.

2Supported by MNZŽS RS

For positive integers n and m the symbol A → (B)_{m}^{n} denotes the statement: For each A ∈ A and for each function f : [A]^{n} → {1, ⋯, m} there are a set B ⊂ A, B ∈ B, and an i ∈ {1, ⋯, m} such that for each Y ∈ [B]^{n}, f(Y) = i. Here [A]^{n} denotes the set of n-element subsets of A. We call f a “coloring”

and say that “B is homogeneous of color i for f”.

This symbol is called the ordinary partition symbol [7]. Several selection principles of the form S_{1}(A, B) have been characterized by the ordinary partition relation (see [7], [4], [6]).

An open cover U of a space X is an ω-cover (resp. k-cover) if X does not belong to U and every finite (resp. compact) subset of X is contained in a member of U. Because we deal with k-covers, we assume that spaces we consider are (infinite) non-compact. An open cover U of X is called a γ-cover [3] if it is infinite and each x ∈ X belongs to all but finitely many elements of U. Notice that it is equivalent to the assertion: Each finite subset of X belongs to all but finitely many members of U. An open cover of a space X is called a γ_{k}-cover of X if each compact subset of X is contained in all but finitely many elements of U and X is not a member of the cover ([5]).

We suppose that all covers are countable. Recall that spaces in which every open k-cover contains a countable k-subcover are called k-Lindelöf.

For a topological space X we denote:

1. Ω – the family of ω-covers of X;

2. K – the family of k-covers of X;

3. Γ – the family of γ-covers of X;

4. Γ_{k} – the family of γ_{k}-covers of X. Let us observe that we have

Γ_{k} ⊂ Γ ⊂ Ω, Γ_{k} ⊂ K ⊂ Ω.

In [3], Gerlits and Nagy introduced the following notion: a space X is a γ-space (or a γ-set) if each ω-cover U of X contains a countable family {U_{n} : n ∈ N} which is a γ-cover of X. They have also proved that the γ-set property of a space X is equivalent to the statement that X satisfies the selection property S_{1}(Ω, Γ). It was shown in [4] that the γ-set property is equivalent also to the selection hypothesis S_{fin}(Ω, Γ).

In [7], it was proved:

Theorem 1. For a space X the following statements are equivalent:

(a) X is a γ-set;

(b) ONE has no winning strategy in the game G_{1}(Ω, Γ) on X;

(c) For all n, m ∈ N, X satisfies Ω → (Γ)_{m}^{n}.

We shall prove here that similar results are true for two recently introduced classes of spaces which are similar to γ-sets. In fact, we give Ramsey theoretical characterizations of those classes of spaces.

Spaces related to γ-sets 127

2. k-γ-sets

In [1], the class of k-γ-sets was introduced as the class of S_{1}(K, Γ)-sets and the following result regarding that class of spaces was shown.

Theorem 2. For a space X the following are equivalent:

(1) X is a k-γ-set;

(2) X satisfies S_{fin}(K, Γ);

(3) ONE has no winning strategy in the game G_{1}(K, Γ) on X.

We show here that this class of spaces also can be described Ramsey-theoreti-cally.

Theorem 3. For a k-Lindelöf space X the following are equivalent:

(1) X is a k-γ-set;

(2) For positive integers n and m, X satisfies K → (Γ)_{m}^{n}.

Proof. We consider the case n = m = 2, because the general case can be easily obtained from it by standard induction arguments.

(1) ⇒ (2): Let U = {U_{1}, U_{2}, ⋯ } be a countable k-cover of X and let f : [U]^{2} → {1, 2} be a coloring. For j ∈ {1, 2} let H_{j} = {V ∈ U : f({U_{1}, V }) = j}. Then at least one of the sets H_{1} and H_{2} is a k-cover of X. Denote such a set by U_{1} and the corresponding j by i_{1}. In a similar way define inductively sets U_{n} of k-covers of X and elements i_{n} from {1, 2} such that

U_{n} = {V ∈ U_{n−1} : f({U_{n}, V }) = i_{n}}.

Apply now the S_{1}(K, Γ) property of X to the sequence (U_{n} : n ∈ N) to choose for each n ∈ N a V_{n} ∈ U_{n} such that V = {V_{n} : n ∈ N} ∈ Γ. Consider now the sets V_{1} := {V_{m} ∈ V : i_{m} = 1} and V_{2} := {V_{m} ∈ V : i_{m} = 2}. At least one of them is infinite and so is a γ-cover of X, as each infinite subset of a γ-cover is also a γ-cover. So, one may suppose that there is an i ∈ {1, 2} satisfying: for each U_{m} ∈ V, i_{m} = i. We have that f({A, B}) = i for each {A, B} ∈ [V]^{2}.

(2) ⇒ (1): Let (U_{n} : n ∈ N) be a sequence of countable k-covers of X and let us suppose that for each n ∈ N, U_{n} = {U_{n,m} : m ∈ N}. Define now V to be the family of all nonempty sets of the form U_{1,n} ∩ U_{n,m}, n, m ∈ N. Clearly, V is a k-cover of X. Let f : [V]^{2} → {1, 2} be define by

f(U_{1,n_{1}} ∩ U_{n_{1},m}, U_{1,n_{2}} ∩ U_{n_{2},l}) = \{ \begin{matrix}1, & if n_{1} = n_{2}, \\ 2, & otherwise. \end{matrix}

Apply K → (Γ)_{2}^{2} to find a γ-cover W ⊂ V and an i ∈ {1, 2} such that whenever U and V are from W, then f({U, V }) = i. Consider two possibilities:

(i) i = 1: Then there is some n ∈ N such that for each W ∈ W we have W ⊂ U_{1,n}. However, this implies that W is not a (γ-) cover of X and this contradiction shows that this case is impossible.

(ii) i = 2: Whenever W ∈ W is of the form U_{1,n} ∩ U_{n,m} choose (one element) H_{n} = U_{n,m} ∈ U_{n}. Otherwise, let H_{n} = ∅. Then the set H := {H_{n} : n ∈ N} is a γ-cover of X (because W refines H), and the sequence (H_{n} : n ∈ N) witnesses for (U_{n} : n ∈ N) that X satisfies S_{fin}(K, Γ). By Theorem 2 it is equivalent to S_{1}(K, Γ), i.e. (1) holds.

3. γ′_{k}-sets

The following class of spaces was introduced in [5]. A space X is said to be a γ′_{k}-set if it satisfies the selection hypothesis S_{1}(K, Γ_{k}).

A characterization of γ′_{k}-sets from [5] is given in the next theorem.

Theorem 4. For a space X the following are equivalent:

(1) X is a γ′_{k}-set;

(2) X satisfies S_{fin}(K, Γ_{k});

(3) ONE does not have a winning strategy in the game G_{1}(K, Γ_{k}) played on X.

We give now a Ramsey-theoretical characterization of γ′_{k}-sets.

Theorem 5. For a k-Lindelöf space X the following are equivalent:

(1) X is a γ′_{k}-set;

(2) For all n, m ∈ N, X satisfies K → (Γ_{k})_{m}^{n}.

Proof. We again consider only the case n = m = 2.

(1) ⇒ (2): We shall use that (1) is equivalent to the fact that ONE has no winning strategy in the game G_{1}(K, Γ_{k}) played on X (Theorem 4).

Suppose U = {U_{1}, U_{2}, ⋯ } is a k-cover of X and let f : [U]^{2} → {1, 2} be a coloring. Let us define a strategy σ for ONE in the game G_{1}(K, Γ_{k}).

In the first round ONE plays σ(∅) = U. Then choose i_{n} ∈ {1, 2}, n ∈ N, such that σ(U_{n}) = {V ∈ U : σ({U_{n}, V }) = i_{n}} is a k-cover of X (see the proof of Theorem 3). Let us write σ(U_{n}) = {U_{n,m} : m ∈ N}. Suppose for each finite sequence (n_{1}, …, n_{p}) of natural numbers we have defined sets U_{n_{1},... ,n_{p}} and i_{n_{1},... ,n_{p−1}} ∈ {1, 2} satisfying the condition {U_{n_{1},... ,n_{p},m} : m ∈ N} is a k-cover of X which is equal to the set

{V ∈ σ(U_{n_{1}}, U_{n_{1},n_{2}}, …, U_{n_{1},n_{2},... ,n_{p}}) : f({U_{n_{1},n_{2},... ,n_{p}}, V }) = i_{n_{1},n_{2},... ,n_{p}}}.

In this way one defines a strategy σ for ONE in G_{1}(K, Γ_{k}). As ONE has no winning strategy, there is a play (for TWO)

U_{n_{1}}, U_{n_{1},n_{2}}, …, U_{n_{1},n_{2},... ,n_{m}} , …

which defeats this strategy. The set {U_{n_{1}}, …, U_{n_{1},n_{2},... ,n_{m}} , … } is a γ_{k}-cover of X. Besides, if p < q, then

f({U_{n_{1},n_{2},... ,n_{p}}, U_{n_{1},n_{2},... ,n_{q}}}) = i_{n_{1},n_{2},... ,n_{p}}.

We may choose i ∈ {1, 2} such that for infinitely many m we have i_{n_{1},n_{2},... ,n_{m}} = i. Then define

V = {U_{n_{1},n_{2},... ,n_{m}} : i_{n_{1},n_{2},... ,n_{m}} = i} ⊂ U.

This set is a γ_{k}-cover of X (because an infinite subset of a γ_{k}-cover is also a γ_{k}-cover) and, by construction, is homogeneous for f of color i.

(2) ⇒ (1): Let (U_{n} : n ∈ N) be a sequence of countable k-covers of X and suppose that for each n, U_{n} = {U_{n;m} : m ∈ N}. Consider now the set V of all nonempty sets of the form U_{1;m} ∩ U_{m;k}, n, k ∈ N. Clearly, V is a k-cover of X. Define f : [V]^{2} → {1, 2} by

f(U_{1;n_{1}} ∩ U_{n_{1};k}, U_{1;n_{2}} ∩ U_{n_{2};m}) = \{ \begin{matrix}1, & if n_{1} = n_{2}, \\ 2, & otherwise. \end{matrix}

Since K → (Γ_{k})_{2}^{2} holds there are j ∈ {1, 2} and a homogeneous for f of color j collection W ⊂ V such that W ∈ Γ_{k}. Consider two possibilities:

(i) j = 1: Then there is some n such that for each W ∈ W we have W ⊂ U_{1,n}. However, this means that W is not a γ_{k}-cover of X and we have a contradiction which shows that this case is impossible.

(ii) j = 2: For each W ∈ W choose, when it is possible, U_{n;k_{n}} to be the second term in the chosen representation of W; otherwise let U_{n;k_{n}} = ∅. Let V′ be the set of all U_{n,k_{n}}’s chosen in this way. Then V′ is a γ_{k}-cover of X witnessing for (U_{n} : n ∈ N) that X satisfies S_{fin}(K, Γ_{k}). Apply now Theorem 4

REFERENCES

[1] G. Di Maio, Lj.D.R. Kočinac and E. Meccariello, Applications of k-covers, Acta Math. Sinica, English Series, to appear.

[2] R. Engelking, General Topology, PWN, Warszawa, 1977.

[3] J. Gerlits and Zs. Nagy, Some properties of C(X), I, Topology Appl. 14 (1982), 151–161.

[4] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, Combinatorics of open covers (II), Topology Appl. 73 (1996), 241–266.

[5] Lj.D.R. Kočinac, γ-sets, γ_{k}-sets and hyperspaces, Math. Balkanica 19, 1–2 (2005), 109–118.

[6] Lj.D.R. Kočinac and M. Scheepers, Combinatorics of open covers (VII): Groupability, Fund. Math. 179 (2003), 131–155.

[7] M. Scheepers, Combinatorics of open covers (I): Ramsey Theory, Topology Appl. 69 (1996), 31–62.

(received 12.11.2005, in revised form 08.03.2006)

Filippo Cammaroto, Dipartimento di Matematica, Università di Messina, 98166 Messina, Italia

E-mail: camfil@unime.it

Ljubiša D.R. Kočinac, Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia

E-mail: lkocinac@ptt.yu