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Topological Games and Hyperspace Topologies

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Abstract

The paper proposes a unified description of hypertopologies, i.e. topologies on the nonempty closed subsets of a topological space, based on the notion of approach spaces introduced by R. Lowen. As a special case of this description we obtain the abstract hit-and-miss, proximal hit-and-miss and a big portion of weak hypertopologies generated by gap and excess functionals (including the Wijsman topology and the finite Hausdorff topology), respectively. We also give characterizations of separation axioms T 0, T 1, T 2, T 3 and complete regularity as well as of metrizability of hypertopologies in this general setting requiring no additional conditions. All this is done to provide a background for proving the main results in Section 4, where we apply topological games (the Banach–Mazur and the strong Choquet game, respectively) to establish various properties of hypertopologies; in particular we characterize Polishness of hypertopologies in this general setting.

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  1. Department of Mathematics and Computer Science, University of North Carolina, Pembroke, NC, 28372, U.S.A.

    László Zsilinszky

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  1. László Zsilinszky
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Zsilinszky, L. Topological Games and Hyperspace Topologies. Set-Valued Analysis 6, 187–207 (1998). https://doi.org/10.1023/A:1008669420995

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