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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The $ \alpha$-boundification of $ \alpha$


Authors: Salvador García-Ferreira and Angel Tamariz-Mascarúa
Journal: Proc. Amer. Math. Soc. 118 (1993), 1301-1311
MSC: Primary 54A25; Secondary 54D30, 54D40
MathSciNet review: 1165054
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Abstract: A space $ X$ is $ < \alpha $-bounded if for all $ A \subseteq X$ with $ \vert A\vert < \alpha $, $ {\operatorname{cl} _X}\;A$ is compact. Let $ B(\alpha )$ be the smallest $ < \alpha $-bounded subspace of $ \beta (\alpha )$ containing $ \alpha $. It is shown that the following properties are equivalent: (a) $ \alpha $ is a singular cardinal; (b) $ B(\alpha )$ is not locally compact; (c) $ B(\alpha )$ is $ \alpha $-pseudocompact; (d) $ B(\alpha )$ is initially $ \alpha $-compact. Define $ {B^0}(\alpha ) = \alpha $ and $ {B^n}(\alpha ) = \{ {\operatorname{cl} _{\beta (\alpha )}}A:A \subseteq {B^{n - 1}}(\alpha ),\vert A\vert < \alpha \} $ for $ 0 < n < \omega $. We also prove that $ {B^2}(\alpha ) \ne {B^3}(\alpha )$ when $ \omega = \operatorname{cf} (\alpha ) < \alpha $. Finally, we calculate the cardinality of $ B(\alpha )$ and prove that, for every singular cardinal $ \alpha ,\;\vert B(\alpha )\vert = \vert B(\alpha ){\vert^\alpha } = \vert N(\alpha ){\vert^{\operatorname{cf} (\alpha )}}$ where $ N(\alpha ) = \{ p \in \beta (\alpha ):\;{\text{there is}}\;A \in p\;{\text{with}}\;\vert A\vert < \alpha \} $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1165054-2
PII: S 0002-9939(1993)1165054-2
Keywords: $ < \alpha $-bounded space, singular cardinal, regular cardinal, $ \alpha $-good point, weak $ {P_\alpha }$-point, $ F$-space, $ \alpha $-pseudocompact, initially $ \alpha $-compact
Article copyright: © Copyright 1993 American Mathematical Society