Baire spaces and infinite games

TL;DR

This paper explores the relationship between Baire spaces and infinite games, specifically the Banach-Mazur game, and establishes the consistency of the converse implication under certain set-theoretic assumptions.

Contribution

It proves the consistency of the converse implication that if a space is Baire in all powers, then the nonempty player has a winning strategy in the Banach-Mazur game, assuming large cardinal axioms.

Findings

The nonempty player's winning strategy implies Baire property in all powers.

The converse implication's consistency is established relative to measurable cardinals.

The results connect infinite game strategies with topological Baire properties.

Abstract

It is well known that if the nonempty player of the Banach-Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box topology. The converse of this implication may be true also: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.