Compactness properties defined by open-point games
Introduction
Definition 1.1 A topological space X is called: sequentially compact if every sequence in X has a convergent subsequence; countably compact if every sequence in X has an accumulation point in X; pseudocompact if every real-valued continuous function defined on X is bounded; selectively sequentially pseudocompact if for every sequence of non-empty open subsets of X, we can choose a point for every in such a way that the sequence has a convergent subsequence; selectively pseudocompact if and only if for every sequence of non-empty open subsets of X, we can choose a point for every in such a way that the sequence 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 chooses a non-empty open subset of X, and Player responds by selecting a point . In the selectively sequentially pseudocompact game on X, Player wins if the sequence has a convergent subsequence; otherwise Player wins. In the selectively pseudocompact game on X, Player wins if the sequence has an accumulation point in X; otherwise Player wins. The (non-)existence of winning strategies for each player in the game (in the game ) 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 has a winning strategy in but does not have a stationary winning strategy even in . (A strategy for Player is called stationary if it depends only on the last move of the opponent, and not on the whole sequence .) This example makes essential use of a van Douwen maximally almost disjoint family of subsets of 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 and give five examples of such properties in Example 2.4. Each topological property 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 on X similar to and in which Player is declared a winner when the sequence selected by satisfies property in X. The arrow (c) from Diagram 2 decomposes into four different arrows – of Diagram 3. In Section 5 we introduce games and as particular cases of the general game .
Theorem 6.1 states that a winning strategy for Player in generates a stationary winning strategy for Player in the game played on a product Z of a space X and the one-point compactification Y of a discrete space of cardinality .
In Section 7, we construct a maximal almost disjoint family consisting of injections from an infinite subset of to a fixed infinite set D built in Theorem 7.1. When , the family is nothing but an “injective version” of a van Douwen MAD family of Raghavan [11, Theorem 2.14]. The family is used in the construction of a locally compact first countable zero-dimensional space X such that Player has a winning strategy in but does not have a stationary winning strategy even in ; see Theorem 8.1. This example is consequently employed in Corollary 8.2 showing that arrow of Diagram 3 is not reversible. In Section 9, we give an example showing that arrow of Diagram 3 is not reversible (Corollary 9.2). The reversibility of arrows and of Diagram 3 remains unclear; see Question 10.1. The reversibility of arrow is equivalent to determinacy of the game .
Section snippets
Topological properties of sequences
For a set X, we identify with the set of all sequences in X.
Definition 2.1 A topological property of sequences is a class is a topological space}, where for every topological space X, such that When , we shall say that the sequence satisfies property in X.
Remark 2.2 Condition (1) means that, for every topological property of sequences , if a sequence of points in a subspace Y of a topological space X satisfies in Y, then it also satisfies 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 be the space of all ordinals less than α with the order topology. The space (is first countable and) has a dense sequentially compact subspace but is not countably compact. Indeed, the sequence does not have an accumulation point in T, so T is not countably compact. Since is sequentially compact, is sequentially compact. Finally, note that D is dense in T
Open-point game
Definition 4.1 Let be a fixed topological property of sequences. For a topological space X, we define the open-point topological game on X between Player and Player as follows. An infinite sequence such that is a non-empty open set and for every is called a play in . Given a play in , we will say that Player wins w if satisfies in X, otherwise, Player wins w.
Notation 4.2 (i) Given a set Y, we use to denote the set of all
Special cases and of the open-point game
Two special cases of the game play a prominent role in this paper.
Definition 5.1 (i) When the topological property of sequences is defined as in item (i) of Example 2.4, we shall denote the game simply by and call it the selectively sequentially pseudocompact game on X. (ii) When the topological property of sequences is defined as in item (v) of Example 2.4, we shall denote the game simply by and call it the selectively pseudocompact game on X.
The abbreviations and
Producing stationary winning strategies for Player in from non-stationary ones
For a set Y, we use to denote the one point compactification of , where 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 in to produce a stationary winning strategy for Player in the game played on the product of X with the one point compactification of the discrete space .
Theorem 6.1 Let 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 , 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 denote the family of all injective functions g from a countably infinite subset of to D. Then there exists a family having two properties: If are distinct, then the set
Example showing that arrow 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 constructed in Theorem 7.1.
Theorem 8.1 There exists a locally compact, first-countable, zero-dimensional space X such that Player has a winning strategy in but does not have a stationary winning strategy even in . Proof Let C be the Cantor set. For every , fix a
Example showing that arrow of Diagram 3 is not reversible
Theorem 9.1 There exists a selectively sequentially pseudocompact space X such that Player has a winning strategy in . 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 , define Note that is homeomorphic to the Cantor set for every . We are going to show that the subspace of has the desired properties. Note
Open questions
We do not know if arrows and of Diagram 3 are reversible.
Question 10.1 Let 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. Is the game determined? Equivalently, is arrow () in Diagram 3 reversible? What can be said for properties from items (i) and (v) of Example 2.4 themselves? Is arrow () in Diagram 3 reversible? What can be said for properties 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 and 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
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The first listed author was supported by CONACYT, México: Estancia Posdoctoral al Extranjero, 178425/277660.