On winning strategies for Banach-Mazur games

TL;DR

This paper formalizes Banach-Mazur games using topological and game theoretic frameworks, providing necessary conditions for players to have winning strategies.

Contribution

It introduces formal definitions and theorems for Banach-Mazur games, establishing foundational results on the existence of winning strategies.

Findings

Provided topological and game theoretic definitions for Banach-Mazur games

Proved necessary conditions for the existence of winning strategies

Formalized the game and its strategic properties

Abstract

We give topological and game theoretic definitions and theorems nec- essary for defining a Banach-Mazur game, and apply these definitions to formalize the game. We then state and prove two theorems which give necessary conditions for existence of winning strategies for players in a Banach-Mazur game.