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Transactions of the American Mathematical Society

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

 

 

A Boolean algebra of regular closed subsets of $ \beta X-X$


Author: R. Grant Woods
Journal: Trans. Amer. Math. Soc. 154 (1971), 23-36
MSC: Primary 54.53
DOI: https://doi.org/10.1090/S0002-9947-1971-0270341-6
MathSciNet review: 0270341
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Abstract: Let X be a locally compact, $ \sigma $-compact, noncompact Hausdorff space. Let $ \beta X$ denote the Stone-Čech compactification of X. Let $ R(X)$ denote the Boolean algebra of all regular closed subsets of the topological space X. We show that the map $ A \to ({\text{cl}_{\beta X}}A) - X$ is a Boolean algebra homomorphism from $ R(X)$ into $ R(\beta X - X)$. Assuming the continuum hypothesis, we show that if X has no more than $ {2^{{\aleph _0}}}$ zero-sets, then the image of a certain dense subalgebra of $ R(X)$ under this homomorphism is isomorphic to the Boolean algebra of all open-and-closed subsets of $ \beta N - N$ (N denotes the countable discrete space). As a corollary, we show that there is a continuous irreducible mapping from $ \beta N - N$ onto $ \beta X - X$. Some theorems on higher-cardinality analogues of Baire spaces are proved, and these theorems are combined with the previous result to show that if S is a locally compact, $ \sigma $-compact noncompact metric space without isolated points, then the set of remote points of $ \beta S$ (i.e. those points of $ \beta S$ that are not in the $ \beta S$-closure of any discrete subspace of S) can be embedded densely in $ \beta N - N$.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0270341-6
Keywords: Stone-Čech compactification, locally compact, $ \sigma $-compact space, regular closed set, Boolean algebra homomorphism, projective cover, irreducible mapping, Stone space, remote points
Article copyright: © Copyright 1971 American Mathematical Society