Infinite games and chain conditions

TL;DR

This paper uses infinite two-person game theory to address classical topology problems, providing new game-theoretic conditions that imply separability and dense $G_\delta$-cover properties in specific spaces.

Contribution

It introduces novel game-theoretic approaches to Suslin's and Arhangel'skii's problems, establishing conditions that guarantee separability and dense covers in topological spaces.

Findings

Linearly ordered spaces satisfying the game-theoretic chain condition are separable.

Compact spaces with the game-theoretic weak Lindelöf property have large $G_\delta$-dense subcollections.

New connections between infinite games and classical topology problems.

Abstract

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on $G_\delta$ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindel\"of property, every cover by $G_\delta$ sets has a continuum-sized subcollection whose union is $G_\delta$-dense.