On the cellularity of 𝛽𝑋-𝑋

Ginsburg, John; Woods, R. Grant

doi:10.1090/S0002-9939-1976-0407789-2

On the cellularity of $\beta X-X$
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by John Ginsburg and R. Grant Woods

Proc. Amer. Math. Soc. 57 (1976), 151-154

DOI: https://doi.org/10.1090/S0002-9939-1976-0407789-2

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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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Cardinal functions in topology

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