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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 $ {2^X}$ the space of closed subsets of X with the finite topology. If X is normal and $ {T_1}$, the map $ F \to {\text{cl}_{\beta X}}F$ is an embedding of $ {2^X}$ onto a dense subspace of $ {2^{\beta X}}$, and, in this way, we regard $ {2^{\beta X}}$ as a compactification of $ {2^X}$. This paper is motivated by the following question. When can $ {2^{\beta X}}$ be identified as the Stone-Čech compactification of $ {2^X}$? In [11], J. Keesling states that $ \beta ({2^X}) = {2^{\beta X}}$ implies $ {2^X}$ is pseudocompact. We give a proof of this result and establish the following partial converse. If $ {2^X} \times {2^X}$ is pseudocompact, then $ \beta ({2^X}) = {2^{\beta X}}$. A corollary of this theorem is that $ \beta ({2^X}) = {2^{\beta X}}$ when X is $ {\aleph _0}$-bounded.

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Keywords: Hyperspace, Stone-Čech compactification, pseudocompactness, topological semilattice
Article copyright: © Copyright 1976 American Mathematical Society