Selection games on continuous functions

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Abstract

In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [12] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on X and Cp(X), the current authors showed similar equivalences in [1] involving the compact subsets of X and Ck(X). By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game equivalence via topological connections, and strengthening of strategies (Pawlikowski and Tkachuk). Moreover, we develop a framework which identifies topological notions to match with generalized versions of the point-open game.

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 Cp(X) 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 Cp(X) [13]. In 2019, Clontz and Holshouser [3] finished this characterization, showing that the discrete selection game on Cp(X) 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 Ck(X), 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,

and use work of Clontz [2] to show that some general classes of games are equivalent.

Section snippets

Definitions and preliminaries

Definition 1

Let X be a space and A(X). We say that A is an ideal-base if, for A1,A2A, there exists A3A so that A1A2A3.

Definition 2

For a topological space X and a collection A(X), we let A¯={clX(A):AA}.

Definition 3

Fix a topological space X and a collection A(X). Then

  • we let Cp(X) denote the set of all continuous functions XR endowed with the topology of point-wise convergence; we also let 0 be the function which identically zero.

  • we let Ck(X) denote the set of all continuous functions XR endowed with the topology of

Strengthening strategies

Lemma 4

Suppose A is an ideal-base, X=A, and let UO(X,A). Then, for each AA, {UU:AU} is infinite. That is, O(X,A)=Λ(X,A).

Proof

Let AA be arbitrary and let U0U be so that AU0. Since XU0, let x1XU0 and let A1A be so that x1A1. Let A1A be so that AA1A1 and let U1U be so that A1U1. Since A1(XU0), we know that U0U1. Inductively continue in this way. 

Corollary 5

Suppose A and B are ideal-bases. Then G1(N[A],¬O(X,B)) is equivalent to G1(N[A],¬Λ(X,B)).

Definition 16

For collections A and B, recall that A refines B

An order on single selection games

Definition 17

Let A, B, C, and D be collections and α be an ordinal. Say that G1α(A,C)IIG1α(B,D) if

  • IImarkG1α(A,C)IImarkG1α(B,D),

  • IIG1α(A,C)IIG1α(B,D),

  • I↑̸G1α(A,C)I↑̸G1α(B,D), and

  • I↑̸preG1α(A,C)I↑̸preG1α(B,D).

Notice that if G1α(A,C)IIG1α(B,D) and G1α(B,D)IIG1α(A,C), then the games are equivalent. Also notice that II is transitive.

Theorem 12

Let A, B, C, and D be collections and α be an ordinal. Suppose there are functions

  • TI,ξ:BA and

  • TII,ξ:A×BB

for each ξα, so that
  • (Tr1)

    If xTI,ξ(B), then TII,ξ(x,B)B

  • (Tr2)

    If xξTI,

Equivalent and dual classes of games

Corollary 14

Let X be a Tychonoff space and A,B(X). Then

  • (i)

    G1(O(X,A),Λ(X,B))IIG1(ΩCA(X),0,ΩCB(X),0),

  • (ii)

    G1(ΩCA(X),0,ΩCB(X),0)IIG1(DCA(X),ΩCB(X),0), and

  • (iii)

    if A consists of closed sets and X is A-normal, thenG1(DCA(X),ΩCB(X),0)IIG1(O(X,A),Λ(X,B)).

Thus if A consists of closed sets and X is A-normal, then the three games are equivalent.

Proof

Let ϕ:CA(X)×ωTX be defined by ϕ(f,n)=f1[(2n,2n)]. Suppose FΩCA(X),0 and let both AA and nω be arbitrary. Choose fF so that f[0;A,2n] and notice that Af1[(2n,2n)].

Covering properties

Lemma 22

Suppose X is a Tychonoff space. Then the following are equivalent:

  • (i)

    IpreG1(NCA(X)(0),¬ΓCB(X),0),

  • (ii)

    IpreG1(NCA(X)(0),¬ΩCB(X),0),

  • (iii)

    cof(NCA(X)(0);NCB(X)(0),)=ω.

Proof

Clearly, (i) implies (ii).

Suppose IpreG1(NCA(X)(0),¬ΩCB(X),0). Then we get a sequence of neighborhoods [0;An,εn]. Now, let BB, ε>0, and consider [0;B,ε]. Suppose [0;An,εn][0;B,ε] for any n. Then we have functions fn[0;An,εn][0;B,ε]. Consider the play [0;A0,ε0],f0, of the game G1(TCA(X),¬ΩCB(X),0) according to the winning strategy.

The main theorems

Theorem 25

Suppose X is a Tychonoff space and A,B(X). Suppose A and B are ideal-bases and that A consists of closed sets. Then the following diagrams are true, where dashed arrows require the assumption that X is A-normal and dotted lines require the assumption that B consists of R-bounded sets.

If X is A-normal, B consists of R-bounded sets, and A consists of Gδ sets, then all of the statements across both diagrams are equivalent.

Proof

Since we have assumed that A and B 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 Gδ sets, A consists of the closed nowhere dense sets, and B is the set of all singleton subsets of X. Then One has a winning strategy in G1(N[A],¬O(X,B)) if and only if X is meager.

Corollary 28

Suppose X is a space, A consists of the Gδ μ-null sets with respect to a Borel measure μ, and B is the set of all singleton subsets of X. Then One has a winning strategy in G1(N[A],¬O(X,B)) if and only if X

Open questions

  • Is there a topological characterization of the statement cof(A;B,)Tωω?

  • Does IG1(KX,ΩX) imply IpreG1(KX,ΩX)?

  • More broadly, to what extent can the Pawlikowski generalization presented here be further generalized?

  • If A is an ideal base, are G1(N[A],¬Γ(X,B)) and G1(N[A],¬O(X,B)) equivalent for player Two?

  • Can the assumption that B consists of R-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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