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# Topological Games and Alster Spaces

Published:2014-02-25
Printed: Dec 2014
• Leandro F. Aurichi,
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil
• Rodrigo R. Dias,
Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo, SP, 05315-970, Brazil
 Format: LaTeX MathJax PDF

## 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 Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.
 Keywords: topological games, selection principles, Alster spaces, Menger spaces, Rothberger spaces, Menger game, Rothberger game, compact-open game, $G_\delta$-topology
 MSC Classifications: 54D20 - Noncompact covering properties (paracompact, Lindelof, etc.) 54G99 - None of the above, but in this section 54A10 - Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

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