The locally finite topology on 2^{𝑋}

Beer, G. A.; Himmelberg, C. J.; Prikry, K.; Van Vleck, F. S.

doi:10.1090/S0002-9939-1987-0897090-2

The locally finite topology on $2^ X$
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by G. A. Beer, C. J. Himmelberg, K. Prikry and F. S. Van Vleck PDF

Proc. Amer. Math. Soc. 101 (1987), 168-172 Request permission

Abstract:

Let $X$ be a metrizable space. A Vietoris-type topology, called the locally finite topology, is defined on the hyperspace ${2^X}$ of all closed, nonempty subsets of $X$. We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on $X$. We also investigate when the locally finite topology coincides with the more usual topologies on ${2^X}$ and when the locally finite topology is metrizable.

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