Selection games on continuous functions
Introduction
The closed discrete selection principle was first studied by Tkachuk in 2017. This property occurs naturally in the course of studying functional analysis. Tkachuk connected this selection principle on with topological properties of X. He then went on to consider the corresponding selection game, creating a partial characterization of winning strategies in that game and finding connections between it, the point-open game on X, and Gruenhage's W-game on [13]. In 2019, Clontz and Holshouser [3] finished this characterization, showing that the discrete selection game on is equivalent to a modification of the point-open game on X. Clontz and Holshouser show this not only for full information strategies but also for limited information strategies.
The current authors continued this work, researching the closed discrete game on , the real-valued continuous functions with the compact open topology [1]. They show that similar connections exist in this setting, with the point-open game on X replaced by the compact-open game. They also isolated general techniques which have use beyond the study of closed discrete selections.
In this paper, we study the problem of closed discrete selection in the general setting of set-open topologies on the space of continuous functions. We use closed discrete selection as a tool not only for comparing X to its space of continuous functions, but also for comparing different set-open topologies to each other. To establish these connections, we prove general statements in three categories:
- 1.
strengthening the strategies in games,
- 2.
criteria for games to be dual,
- 3.
characterizations of strong strategies in abstract point-open games,
Section snippets
Definitions and preliminaries
Definition 1 Let X be a space and . We say that is an ideal-base if, for , there exists so that .
Definition 2 For a topological space X and a collection , we let .
Definition 3 Fix a topological space X and a collection . Then we let denote the set of all continuous functions endowed with the topology of point-wise convergence; we also let 0 be the function which identically zero. we let denote the set of all continuous functions endowed with the topology of
Strengthening strategies
Lemma 4 Suppose is an ideal-base, , and let . Then, for each , is infinite. That is, . Proof Let be arbitrary and let be so that . Since , let and let be so that . Let be so that and let be so that . Since , we know that . Inductively continue in this way. □
Corollary 5 Suppose and are ideal-bases. Then is equivalent to .
Definition 16 For collections and , recall that refines
An order on single selection games
Definition 17 Let , , , and be collections and α be an ordinal. Say that if , , , and .
Notice that if and , then the games are equivalent. Also notice that is transitive.
Theorem 12 Let , , , and be collections and α be an ordinal. Suppose there are functions and If , then If
for each , so that
Equivalent and dual classes of games
Corollary 14 Let X be a Tychonoff space and . Then , , and if consists of closed sets and X is -normal, then
Thus if consists of closed sets and X is -normal, then the three games are equivalent.
Proof
Let be defined by . Suppose and let both and be arbitrary. Choose so that and notice that .
Covering properties
Lemma 22 Suppose X is a Tychonoff space. Then the following are equivalent: , , .
Proof
Clearly, (i) implies (ii).
Suppose . Then we get a sequence of neighborhoods . Now, let , , and consider . Suppose for any n. Then we have functions . Consider the play of the game according to the winning strategy.
The main theorems
Theorem 25 Suppose X is a Tychonoff space and . Suppose and are ideal-bases and that consists of closed sets. Then the following diagrams are true, where dashed arrows require the assumption that X is -normal and dotted lines require the assumption that consists of -bounded sets. If X is -normal, consists of -bounded sets, and consists of sets, then all of the statements across both diagrams are equivalent. Proof Since we have assumed that and are ideal-bases, Lemma 11 implies that
Applications
Corollary 27, Corollary 28 are direct applications of Lemma 24. Corollary 27 Suppose X is a space where all closed sets are sets, consists of the closed nowhere dense sets, and is the set of all singleton subsets of X. Then One has a winning strategy in if and only if X is meager. Corollary 28 Suppose X is a space, consists of the μ-null sets with respect to a Borel measure μ, and is the set of all singleton subsets of X. Then One has a winning strategy in if and only if X
Open questions
- •
Is there a topological characterization of the statement ?
- •
Does imply ?
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More broadly, to what extent can the Pawlikowski generalization presented here be further generalized?
- •
If is an ideal base, are and equivalent for player Two?
- •
Can the assumption that consists of -bounded sets be removed from Theorem 25, Theorem 26?
- •
To what extent can the techniques in this paper be used to study more complex selection principles like
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