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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the cellularity of $\beta X-X$

Authors: John Ginsburg and R. Grant Woods
Journal: Proc. Amer. Math. Soc. 57 (1976), 151-154
MSC: Primary 54A25; Secondary 54D40
MathSciNet review: 0407789
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Abstract: For a topological space $X$, let $c(X)$ denote the cellularity of $X$, and let $k(X)$ denote the least cardinal of a cobase for the compact subsets of $X$. It is shown that, if $X$ is a completely regular Hausdorff space, $c(\beta X - X) \leqslant {2^{c(X)k(X)}}$, and examples are given to show that this inequality is sharp. It is also shown that if $X$ is an extremally disconnected completely regular Hausdorff space for which $c(\beta X - X) > {2^{k(X)}}$, then $\beta X - X$ contains a discrete ${C^ \ast }$-embedded subspace of cardinality $k{(X)^ + }$.

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Keywords: Stone-Čech compactification, cellularity, cobase for the compact sets, extremally disconnected space
Article copyright: © Copyright 1976 American Mathematical Society