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Topological games and Alster spaces

  • 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
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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 topological games, selection principles, Alster spaces, Menger spaces, Rothberger spaces, Menger game, Rothberger game, compact-open game, $G_\delta$-topology
MSC Classifications: 54D20, 54G99, 54A10 show english descriptions Noncompact covering properties (paracompact, Lindelof, etc.)
None of the above, but in this section
Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
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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