Some observations on the mildly Menger property and topological games

TL;DR

This paper introduces new topological games related to the mildly Menger property, explores their equivalences in zero-dimensional spaces, and provides conditions under which players have winning strategies.

Contribution

It defines new games, establishes their equivalences, and proves strategic results for spaces with certain decompositions.

Findings

Mildly Menger game is equivalent to the Menger game in zero-dimensional spaces.

The compact-clopen game is equivalent to the compact-open game in zero-dimensional spaces.

If a space is a union of countably many quasi-components of compact sets, then TWO has a winning strategy in the mildly Menger game.

Abstract

In this paper, we defined two new games - the mildly Menger game and the compact-clopen game. In a zero-dimensional space, the Menger game is equivalent to the mildly Menger game and the compact-open game is equivalent to the compact-clopen game. An example is given for a space on which the mildly Menger game is undetermined. Also we introduced a new game namely K-quasi-component-clopen game and proved that this game is equivalent to the compact-clopen game. Then we proved that if a topological space is a union of countably many quasi-components of compact sets, then TWO has a winning strategy in the mildly Menger game.