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Proceedings of the American Mathematical Society

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On barely $ \alpha$-compact spaces and remote points in $ \beta\sb \alpha X\sb X$


Authors: Robert L. Blair, Lech T. Polkowski and Mary Anne Swardson
Journal: Proc. Amer. Math. Soc. 107 (1989), 1079-1085
MSC: Primary 54D40; Secondary 54A25, 54D60, 54G05
MathSciNet review: 1019758
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Abstract: Let $ X$ be a Tychonoff space. It was proved by Terada that if the cellularity of $ X$ is not Ulam-measurable, then no point of $ \upsilon X\backslash X$ is a remote point of $ X$. In this paper we generalize this result by proving that if $ X$ is barely $ \alpha $-compact, then no point in $ {\beta _\alpha }X\backslash X$ is an $ \alpha $-exotic point of $ X$. This implies, in particular, that if no cellular family in $ X$ has a $ \alpha $-measurable cardinality, then no point of $ {\beta _\alpha }X\backslash X$ is a remote point of $ X$; the Terada theorem then follows as a corollary when $ \alpha = {\omega _1}$.


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DOI: https://doi.org/10.1090/S0002-9939-1989-1019758-1
Keywords: Čech-Stone compactification, $ \alpha $-compactification, realcompactification, barely $ \alpha $-compact space, remote point, $ \alpha $-exotic point, regular closed set, cellular family, measurable cardinal, absolute
Article copyright: © Copyright 1989 American Mathematical Society