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Proceedings of the American Mathematical Society

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

 

 

A characterization of a space with countable infinity


Author: Akihiro Okuyama
Journal: Proc. Amer. Math. Soc. 28 (1971), 595-597
MSC: Primary 54.53
DOI: https://doi.org/10.1090/S0002-9939-1971-0276929-6
MathSciNet review: 0276929
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Abstract: It is well known that, for a countable discrete space $ N$,

$\displaystyle \vert\beta N - N\vert = {2^{{2^{{\aleph _0}}}}}.$

So, any completely regular $ {T_1}$ space $ X$ with $ \beta X - X\vert \leqq {\aleph _0}$ does not contain any infinite discrete subspace. In this paper, we characterize those completely regular $ {T_1}$ spaces with countable infinity as follows: Such a space $ X$ is characterized by the two properties.

(a) $ X$ is pseudocompact.

(b) There exist a compact metric space $ Y$ and a continuous map $ f$ from $ X$ onto $ Y$ so that the subset $ {Y_0} = \{ y:{f^{ - 1}}(y)$ of $ Y$ is countable and $ c{l_{\beta X}}{f^{ - 1}}(y) - {f^{ - 1}}(y)$ is one point whenever $ y \in {Y_0}$. (In particular, for any $ y$ in $ {Y_0},\beta ({f^{ - 1}}(y)) - {f^{ - 1}}(y)$ is one point if $ X$ is normal.)


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DOI: https://doi.org/10.1090/S0002-9939-1971-0276929-6
Keywords: Stone-Čech compactification, countable infinity, compact metric space, pseudocompact space
Article copyright: © Copyright 1971 American Mathematical Society