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Keywords:
Menger property; Menger game; $\sigma$-compact spaces; limited information strategies
Summary:
As shown by Telgársky and Scheepers, winning strategies in the Menger game characterize $\sigma$-compactness amongst metrizable spaces. This is improved by showing that winning Markov strategies in the Menger game characterize $\sigma$-compactness amongst regular spaces, and that winning strategies may be improved to winning Markov strategies in second-countable spaces. An investigation of 2-Markov strategies introduces a new topological property between $\sigma$-compact and Menger spaces.
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