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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Clopen sets in hyperspaces


Author: Paul Bankston
Journal: Proc. Amer. Math. Soc. 54 (1976), 298-302
MSC: Primary 54B20; Secondary 06A20
MathSciNet review: 0405332
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Abstract: Let $ X$ be a space and let $ H(X)$ denote its hyperspace (= all nonempty closed subsets of $ X$ topologized via the Vietoris topology). Then $ X$ is Boolean (= totally disconnected compact Hausdorff) iff $ H(X)$ is Boolean; and if $ B$ denotes the characteristic algebra of clopen sets in $ X$ then the corresponding algebra for $ H(X)$ is the free algebra generated by $ B$ modulo the ideal which ``remembers'' the upper semilattice structure of $ B$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0405332-5
PII: S 0002-9939(1976)0405332-5
Keywords: Hyperspace, Boolean space, Boolean algebra, clopen sets, characteristic algebra, Stone space
Article copyright: © Copyright 1976 American Mathematical Society