Topological games and Alster spaces

TL;DR

This paper explores the relationships between topological games like Rothberger, Menger, and compact-open, and their connection to properties involving G_delta covers, revealing new implications for Alster spaces and G_delta-topologies.

Contribution

It establishes new links between winning strategies in topological games and properties of Alster spaces and G_delta-topologies, expanding understanding of these concepts.

Findings

Winning strategies in the Menger game imply the space is Alster.

Winning strategies in the Rothberger game imply the G_delta-topology is Lindelof.

The Menger and compact-open games are not dual under certain conditions.

Abstract

In this paper we study connections between topological games such as Rothberger, Menger and compact-open, and relate these games to properties involving covers by G_{\delta} subsets. The results include: (1) If Two has a winning strategy in the Menger game on a regular space X, then X is an Alster space. (2) If Two has a winning strategy in the Rothberger game on a topological space X, then the G_\delta-topology on X is Lindelof. (3) The Menger game and the compact-open game are (consistently) not dual.