The Banach–Mazur game and domain theory

We prove that player α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} has a winning strategy in the Banach–Mazur game on a space X if and only if X is F-Y countably π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}-domain representable. We show that Choquet complete spaces are F-Y countably domain representable. We give an example of a space, which is F-Y countably domain representable, but which is not F-Y π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}-domain representable.


Introduction.
The famous Banach-Mazur game was invented by Mazur in 1935. For the history of game theory and facts about game theory, the reader is referred to the survey [12]. Let X be a topological space and X = A ∪ B be any given decomposition of X into two disjoint sets. The game BM (X, A, B) is played as follows: Two players, named α and β, alternately choose open nonempty sets with We study a well-known modification of this game considered by Choquet in 1958, known as Banach-Mazur game or Choquet game. Player α and β alternately choose open nonempty sets with U 0 ⊇ V 0 ⊇ U 1 ⊇ V 1 · · · . In the first round, player β starts by choosing a nonempty open set U 0 .
Player α wins this play if n∈ω V n = ∅. Otherwise β wins. Denote this game by BM (X). Every finite sequence of sets (U 0 , . . . , U n ), obtained by the first n steps in this game is called partial play of β. A strategy for player α in the game BM (X) is a map s that assigns to each partial play (U 0 , . . . , U n ) of β a nonempty open set V n ⊆ U n . The strategy s is called a winning strategy for player α if player α always wins the play of the game using this strategy. The space X is called weakly α-favorable (see [13]) if X admits a winning strategy for player α in the game BM (X). We say that a partial play The strong Choquet game is defined as follows: In the n-th round, player β selects a point x n and an open set U n such that x n ∈ U n ⊆ V n−1 and α responds with an open set V n such that x n ∈ V n ⊆ U n . Player α wins if n∈ω V n = ∅. Otherwise β wins. We say that a partial play (W 0 , x 0 , . . . , W k , x k ) is stronger than (U 0 , y 0 , . . . , U m , y m ) if m ≤ k and U 0 =W 0 , . . . , U m = W m and x 0 = y 0 , . . . , x m = y m . We denote this by (U 0 , y 0 , . . . , U m , y m ) (W 0 , x 0 , . . . , W k , x k ). We denote a sequence (W 0 , x 0 , . . . , W k , x k ) by ( − → x • − → W )(k). A topological space X is called Choquet complete if player α has a winning strategy in the strong Choquet game, and we then write Ch(X).
For a topological space X, let τ (X) denote the topology on the set X and τ * (X) = τ (X)\{∅}. A family P of open nonempty sets is called a π-base if for every open nonempty set U , there is P ∈ P such that P ⊆ U.
A dcpo (directed complete partial order) is a poset (P, ) in which every directed set has a supremum. If p, q ∈ P , then we say that "p is far below q" whenever for any directed set D with q A domain is a dcpo in which every element q is the supremum of the directed set {p ∈ P : "p is far below q"}. This notion has been introduced by D. Scott as a model for the λ-calculus, for more information see [1], [10]. Domain representable topological spaces were introduced by Bennett and Lutzer [2]. We say that a topological space is domain representable if it is homeomorphic to the space of maximal elements of some domain topologized with the Scott topology. In 2013, Fleissner and Yengulalp [3] introduced an equivalent definition of a domain representable space for T 1 topological spaces. We do not assume the antisymmetry condition on the relation . AsÖnal and Vural suggested in [11], if we need an additional antisymmetric property, let us consider the equivalent relation E on the set Q defined by "pEq if and only if (p q and q p) or p = q". We do not assume any separation axioms, if it is not explicitly stated.
We say that a topological space In [4], Fleissner and Yengulalp introduced the notion of a π-domain representable space, as this is analogous to the notion of a domain representable space.
We say that a topological space 2. π-domain representable spaces. In [5], Kenderov and Revalski have shown that the set E = {f ∈ C(X) : f attains its minimum in X} contains a G δ dense subset of C(X) is equivalent to the existence of a winning strategy for player α in the Banach-Mazur game. Oxtoby [9] showed that if X is a metrizable space, then player α has a winning strategy in BM (X) if and only if X contains a dense completely metrizable subspace. Krawczyk and Kubiś [6] have characterized the existence of winning strategies for player α in the abstract Banach-Mazur game played with finitely generated structures instead of open sets. In [7], there has been presented a version of the Banach-Mazur game played on a partially ordered set. We give a characterization of the existence of a winning strategy for player α in the Banach-Mazur game using the notion "π-domain representable space" introduced by W. Fleissner and L. Yengulalp.

Theorem 1. A topological space X is weakly α-favorable if and only if X is F-Y countably π-domain representable.
Proof. If X is F-Y countably π-domain representable, then it is easy to show that X is weakly α-favorable.
Assume that X is weakly α-favorable. We shall show that X is F-Y countably π-domain representable. Let s be a winning strategy for player α in BM (X). We consider a family Q consisting of all finite sequences is a partial play and m ≤ i, i.e., . Let us define a relation on the family Q: Since is transitive, is transitive. Let us define a map B : Q → τ * (X) by the formula Since {s(V ) : V ∈ τ * (X)} is a π-base, {B(q) : q ∈ Q} is a π-base for τ . It is easy to see that the map B satisfies the condition (πD3).
Towards item (πD4), let p, q ∈ Q be such that B(q) ∩ B(p) = ∅ and ) and s is a winning strategy, we find an element Step by step we find a partial play . Similarly, as for the sequence p, for the sequence q, we define Continuing in this way, we get an element . . . , − → W k (l k ) such that p, q r and r ∈ Q.
Next we show the condition (πD5 ω1 ). Let D ⊆ Q be a countable upward directed set and let D = {p n : n ∈ ω}. We define a chain {q n : n ∈ ω} ⊆ D ⊆ Q such that p n q n for n ∈ ω. By the condition (πD3), we get {B(q n ) : n ∈ ω} ⊆ {B(p) : p ∈ D}. Each q n ∈ Q is of the form q n = − → W n 0 (l n 0 ), . . . , − → W n kn (l n kn ) .
Vol. 114 (2020) The Banach-Mazur game and domain theory 55 ). Inductively, we can choose a sequence {s( − → U n (l n jn )) : n ∈ ω} such that . Since s is a winning strategy for player α, we have We give an example of a space, which is F-Y countably domain representable, but which is not F-Y π-domain representable. Note that this space is F-Y countably π-domain representable and not F-Y domain representable. Example 1. We consider the space where supp x = {α ∈ ω 1 : x(α) = 1} for x ∈ {0, 1} ω1 , with the topology (ω 1 -box topology) generated by the base We shall define a triple (Q, , B). Let Q = B, and the map B : Q → Q be the identity. Define a relation in the following way: It is easy to see that the relation is transitive and that it satisfies the condition (D3). Now, we prove the condition (D4). Let x ∈ X and pr −1 . We prove the condition (D5 ω1 ). Let D ⊆ B be a countable upward directed family. We can construct a chain {pr −1 An (x An ) : n ∈ ω} ⊆ D such that for each set pr −1 A (x A ) ∈ D, there exists n ∈ ω such that pr −1 A (x A ) pr −1 An (x An ). Let B = {A n : n ∈ ω}. Since {pr −1 An (x An ) : n ∈ ω} is a chain, there is x B ∈ {0, 1} B such that x B A n = x An for n ∈ ω. Then An (x An ) : n ∈ ω} = pr −1 B (x B ) ∈ B, and pr −1 B (x B ) ⊆ D. This completes the proof that the space σ {0, 1} ω1 is F-Y countably domain representable. Now we show that X = σ {0, 1} ω1 is not F-Y π-domain representable. Suppose that there exists a triple (Q, , B) satisfying the conditions (πD1)-(πD5). The family P = {B(q) : q ∈ Q} is a π-base. By induction, we define a sequence {Q α : α < ω 1 } such that the following conditions are satisfied: (1) Q α ∈ [Q] ≤ω and Q α is upward directed, for α < ω 1 , (2) {B(q) : q ∈ Q α } = pr −1 Aα (x Aα ) ∈ B for some A α ∈ [ω 1 ] ≤ω and some x Aα ∈ {0, 1} Aα , for α < ω 1 ,