The Banach--Mazur game and the strong Choquet game in domain theory

TL;DR

This paper characterizes when a player has a winning strategy in the Banach--Mazur game using F-Y countably domain representability and explores the relationship with Choquet completeness in domain theory.

Contribution

It establishes a precise equivalence between winning strategies in the Banach--Mazur game and F-Y countably $\pi$-domain representability, and relates Choquet completeness to F-Y countably domain representability.

Findings

Player $\alpha$ has a winning strategy iff $X$ is F-Y countably $\pi$-domain representable.

Choquet complete spaces are F-Y countably domain representable.

Existence of a space that is F-Y countably domain representable but not F-Y $\pi$-domain representable.

Abstract

We prove that a player $\alpha$ has a winning strategy in the Banach--Mazur game on a space $X$ if and only if $X$ is F-Y countably $\pi$-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 it is not F-Y $\pi$-domain representable.