Equivalence of the Rothberger and $2$-Rothberger Games for Hausdorff Spaces

TL;DR

This paper proves that in Hausdorff spaces, the Rothberger game, the 2-Rothberger game, and a restricted Menger game are all equivalent, clarifying their relationships and answering a previously posed question.

Contribution

It establishes the equivalence of Rothberger, 2-Rothberger, and a restricted Menger game in Hausdorff spaces, extending understanding of these topological game relationships.

Findings

Rothberger and 2-Rothberger games are equivalent in Hausdorff spaces.

These games are also equivalent to a restricted Menger game.

The results answer a question by Aurichi, Bella, and Dias.

Abstract

We prove that in any Hausdorff space, the Rothberger game is equivalent to the $k$-Rothberger game, i.e. the game in which player II chooses $k$ open sets in each move. This result follows from a more general theorem in which we show these games are equivalent to a game we call the restricted Menger game. In this game I knows immediately in advance of playing each open cover how many open sets II will choose from that open cover. This result illuminates the relationship between the Rothberger and Menger games in Hausdorff spaces. The equivalence of these games answers a question posed by Aurichi, Bella, and Dias, at least in the context of Hausdorff spaces.