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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
DOI: https://doi.org/10.1090/S0002-9939-1976-0407789-2
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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DOI: https://doi.org/10.1090/S0002-9939-1976-0407789-2
Keywords: Stone-Čech compactification, cellularity, cobase for the compact sets, extremally disconnected space
Article copyright: © Copyright 1976 American Mathematical Society