On the Stone-Čech compactification of the space of closed sets

Author:
John Ginsburg

Journal:
Trans. Amer. Math. Soc. **215** (1976), 301-311

MSC:
Primary 54B20

MathSciNet review:
0390992

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Abstract: For a topological space *X*, we denote by the space of closed subsets of *X* with the finite topology. If *X* is normal and , the map is an embedding of onto a dense subspace of , and, in this way, we regard as a compactification of . This paper is motivated by the following question. When can be identified as the Stone-Čech compactification of ? In [11], J. Keesling states that implies is pseudocompact. We give a proof of this result and establish the following partial converse. If is pseudocompact, then . A corollary of this theorem is that when *X* is -bounded.

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DOI:
https://doi.org/10.1090/S0002-9947-1976-0390992-2

Keywords:
Hyperspace,
Stone-Čech compactification,
pseudocompactness,
topological semilattice

Article copyright:
© Copyright 1976
American Mathematical Society