Selection Principles and Baire spaces

TL;DR

This paper investigates the conditions under which the Banach-Mazur game on a separable metric space is determined, focusing on the Hurewicz covering property and contrasting it with the Menger property.

Contribution

It establishes that the Banach-Mazur game is determined on spaces with the Hurewicz property, highlighting a specific topological condition affecting game determinacy.

Findings

Banach-Mazur game is determined on spaces with Hurewicz property

Determinacy does not hold when replacing Hurewicz with Menger property

Provides insight into the relationship between covering properties and game theory in topology

Abstract

We prove that if X is a separable metric space with the Hurewicz covering property, then the Banach-Mazur game played on X is determined. The implication is not true when "Hurewicz covering property" is replaced with "Menger covering property".