Compactness properties defined by open-point games

Dedicated to Professor Alexander V. Arhangel'skiĭ on the occasion of his 80th anniversary
https://doi.org/10.1016/j.topol.2020.107196Get rights and content

Abstract

Let S be a topological property of sequences (such as, for example, “to contain a convergent subsequence” or “to have an accumulation point”). We introduce the following open-point game OP(X,S) on a topological space X. In the nth move, Player O chooses a non-empty open set UnX, and Player P responds by selecting a point xnUn. Player P wins the game if the sequence {xn:nN} satisfies property S in X; otherwise, Player O wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We thoroughly investigate these properties and construct examples distinguishing half of them, for an arbitrary property S sandwiched between sequential compactness and countable compactness.

Introduction

Definition 1.1

A topological space X is called:

  • (i)

    sequentially compact if every sequence in X has a convergent subsequence;

  • (ii)

    countably compact if every sequence in X has an accumulation point in X;

  • (iii)

    pseudocompact if every real-valued continuous function defined on X is bounded;

  • (iv)

    selectively sequentially pseudocompact if for every sequence {Un:nN} of non-empty open subsets of X, we can choose a point xnUn for every nN in such a way that the sequence {xn:nN} has a convergent subsequence;

  • (v)

    selectively pseudocompact if and only if for every sequence {Un:nN} of non-empty open subsets of X, we can choose a point xnUn for every nN in such a way that the sequence {xn:nN} has an accumulation point in X.

The properties (i)–(iii) are well known [6], while selective properties (iv) and (v) were introduced recently in [3]. It was proved in [3, Theorem 2.1] that the property from item (v) is equivalent to the notion of strong pseudocompactness introduced earlier in [7].

Diagram 1 summarizes relations between the properties from the above definition.

None of the arrows in Diagram 1 are reversible; see [3].

Selective sequential pseudocompactness is the only property in Diagram 1 which is fully productive; that is, any Tychonoff product of selectively sequentially pseudocompact spaces is selectively sequentially pseudocompact [3]. This productivity is a major advantage of selective sequential pseudocompactness when compared to other compactness-like properties.

In this paper, we define two open-point topological games closely related to the class of selectively (sequentially) pseudocompact spaces. Let X be a topological space. At round n, Player O chooses a non-empty open subset Un of X, and Player P responds by selecting a point xnUn. In the selectively sequentially pseudocompact game Ssp(X) on X, Player P wins if the sequence {xn:nN} has a convergent subsequence; otherwise Player O wins. In the selectively pseudocompact game Sp(X) on X, Player P wins if the sequence {xn:nN} has an accumulation point in X; otherwise Player O wins. The (non-)existence of winning strategies for each player in the game Ssp(X) (in the game Sp(X)) defines a compactness-like property of X sandwiched between sequential compactness (countable compactness) and selective sequential pseudocompactness (selective pseudocompactness) of X. In this way we develop a fine structure of the area represented by arrows 1 and 2 of Diagram 1. Furthermore, we construct examples showing that the newly introduced notions are mostly distinct. The most sophisticated example is locally compact, first-countable, zero-dimensional space X such that Player P has a winning strategy in Ssp(X) but does not have a stationary winning strategy even in Sp(X). (A strategy for Player P is called stationary if it depends only on the last move Un of the opponent, and not on the whole sequence (U1,U2,,Un).) This example makes essential use of a van Douwen maximally almost disjoint family of subsets of N due to Raghavan [11, Theorem 2.14].

The paper is organized as follows. In Section 2, we introduce the general notion of a topological property of sequences S and give five examples of such properties in Example 2.4. Each topological property S of sequences gives rise to four natural properties of topological spaces; see Definition 2.3 and Diagram 2. In Section 3 we show that arrows (a) and (b) in this diagram are not reversible (Example 3.3). In order to show that arrow (c) is not reversible either, in Section 4 we introduce a general open-point game OP(X,S) on X similar to Ssp(X) and Sp(X) in which Player P is declared a winner when the sequence {xn:nN} selected by P satisfies property S in X. The arrow (c) from Diagram 2 decomposes into four different arrows (c1)(c4) of Diagram 3. In Section 5 we introduce games Ssp(X) and Sp(X) as particular cases of the general game OP(X,S).

Theorem 6.1 states that a winning strategy for Player O in OP(X,S) generates a stationary winning strategy for Player O in the game OP(Z,S) played on a product Z of a space X and the one-point compactification Y of a discrete space of cardinality |X|.

In Section 7, we construct a maximal almost disjoint family F consisting of injections from an infinite subset of N to a fixed infinite set D built in Theorem 7.1. When D=N, the family F is nothing but an “injective version” of a van Douwen MAD family of Raghavan [11, Theorem 2.14]. The family F is used in the construction of a locally compact first countable zero-dimensional space X such that Player P has a winning strategy in Ssp(X) but does not have a stationary winning strategy even in Sp(X); see Theorem 8.1. This example is consequently employed in Corollary 8.2 showing that arrow (c1) of Diagram 3 is not reversible. In Section 9, we give an example showing that arrow (c4) of Diagram 3 is not reversible (Corollary 9.2). The reversibility of arrows (c2) and (c3) of Diagram 3 remains unclear; see Question 10.1. The reversibility of arrow (c2) is equivalent to determinacy of the game OP(X,S).

Section snippets

Topological properties of sequences

For a set X, we identify XN with the set of all sequences {xn:nN} in X.

Definition 2.1

A topological property of sequences is a class S={SX:X is a topological space}, where SXXN for every topological space X, such thatSYSX whenever Y is a subspace of X.

When sSX, we shall say that the sequence sXN satisfies property S in X.

Remark 2.2

Condition (1) means that, for every topological property of sequences S, if a sequence of points in a subspace Y of a topological space X satisfies S in Y, then it also satisfies S in X.

Definition 2.3

On reversibility of arrows in Diagram 2

Let us recall two examples from the literature.

Example 3.1

Let α be an ordinal and [0,α) be the space of all ordinals less than α with the order topology. The space T=[0,ω+1)×[0,ω1+1){(ω,ω1)} (is first countable and) has a dense sequentially compact subspace but is not countably compact. Indeed, the sequence {(n,ω1):nN} does not have an accumulation point in T, so T is not countably compact. Since [0,ω+1) is sequentially compact, D=[0,ω+1)×[0,ω1) is sequentially compact. Finally, note that D is dense in T

Open-point game OP(X,S)

Definition 4.1

Let S be a fixed topological property of sequences. For a topological space X, we define the open-point topological game OP(X,S) on X between Player O and Player P as follows. An infinite sequence w=(U1,x1,U2,x2,) such that Un is a non-empty open set and xnUn for every nN is called a play in OP(X,S). Given a play w=(U1,x1,U2,x2,) in OP(X,S), we will say that Player P wins w if {xn:nN} satisfies S in X, otherwise, Player O wins w.

Notation 4.2

(i) Given a set Y, we use Seq(Y) to denote the set of all

Special cases Ssp(X) and Sp(X) of the open-point game OP(X,S)

Two special cases of the game OP(X,S) play a prominent role in this paper.

Definition 5.1

(i) When the topological property of sequences S is defined as in item (i) of Example 2.4, we shall denote the game OP(X,S) simply by Ssp(X) and call it the selectively sequentially pseudocompact game on X.

(ii) When the topological property of sequences S is defined as in item (v) of Example 2.4, we shall denote the game OP(X,S) simply by Sp(X) and call it the selectively pseudocompact game on X.

The abbreviations and

Producing stationary winning strategies for Player O in OP(X,S) from non-stationary ones

For a set Y, we use α(Ydisc) to denote the one point compactification of Ydisc, where Ydisc is the set Y endowed with the discrete topology.

In the following theorem, we describe a general technique which employs a winning strategy for Player O in OP(X,S) to produce a stationary winning strategy for Player O in the game OP(X×α(Seq(X)disc),S) played on the product of X with the one point compactification of the discrete space Seq(X)disc.

Theorem 6.1

Let S be a topological property of sequences preserved by

“Injective version” of a van Douwen MAD family on an arbitrary set

In our construction of an example in the next section, we shall need the following set-theoretic result of independent interest. When D=N, this result becomes an “injective version” of a van Douwen MAD family constructed by Raghavan in [11, Theorem 2.14].

Theorem 7.1

Let D be an infinite set and let I(D) denote the family of all injective functions g from a countably infinite subset dom(g) of N to D. Then there exists a family FI(D) having two properties:

  • (A)

    If f,gF are distinct, then the set {ndom(f)dom(g):

Example showing that arrow (c1) of Diagram 3 is not reversible

In [1, Section 5], Berner gave an example of a pseudocompact space without a dense relatively countably compact subspace. The space from our next theorem is a quite significant modification of Berner's example based on the family F constructed in Theorem 7.1.

Theorem 8.1

There exists a locally compact, first-countable, zero-dimensional space X such that Player P has a winning strategy in Ssp(X) but does not have a stationary winning strategy even in Sp(X).

Proof

Let C be the Cantor set. For every cC, fix a

Example showing that arrow (c4) of Diagram 3 is not reversible

Theorem 9.1

There exists a selectively sequentially pseudocompact space X such that Player O has a winning strategy in Sp(X).

Proof

Berner constructed a pseudocompact space X that does not contain a dense relatively countably compact subspace; see [1, Section 3]. Let us describe this space. For every αω1, defineXα={x2ω1:x(α)=1 and x(γ)=0 for all γω1 with γ>α}. Note that Xα is homeomorphic to the Cantor set 2ω for every αω.

We are going to show that the subspaceX=α<ω1Xα of 2ω1 has the desired properties.

Note

Open questions

We do not know if arrows (c2) and (c3) of Diagram 3 are reversible.

Question 10.1

Let S be a topological property of sequences weaker than the property from item (i) and stronger than the property from item (v) of Example 2.4.

  • (i)

    Is the game OP(X,S) determined? Equivalently, is arrow (c2) in Diagram 3 reversible? What can be said for properties S from items (i) and (v) of Example 2.4 themselves?

  • (ii)

    Is arrow (c3) in Diagram 3 reversible? What can be said for properties S from items (i) and (v) of Example 2.4

Acknowledgements

We are grateful to Professor Franklin Tall for his kind suggestion to consider a game-theoretic version of our selective sequential pseudocompactness property from [3] during his visit to Ehime University in December 2016. It is this suggestion which led us to the introduction of games Ssp(X) and Sp(X) from Section 5.

The second listed author would like to thank cordially Professor Yasushi Hirata for answering his question raised at the talk at Yokohama Topology Seminar on October 27, 2017 and

References (12)

There are more references available in the full text version of this article.

Cited by (1)

  • A point-picking game

    2024, Topology and its Applications
1

The first listed author was supported by CONACYT, México: Estancia Posdoctoral al Extranjero, 178425/277660.

2

The second listed author was partially supported by the Grant-in-Aid for Scientific Research (C) No. 26400091 by the Japan Society for the Promotion of Science (JSPS).

View full text