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

Abstract:

A space $X$ is $< \alpha$-bounded if for all $A \subseteq X$ with $|A| < \alpha$, ${\operatorname {cl} _X}\;A$ 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 ${B^n}(\alpha ) = \{ {\operatorname {cl} _{\beta (\alpha )}}A:A \subseteq {B^{n - 1}}(\alpha ),|A| < \alpha \}$ 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 $\alpha ,\;|B(\alpha )| = |B(\alpha ){|^\alpha } = |N(\alpha ){|^{\operatorname {cf} (\alpha )}}$ where $N(\alpha ) = \{ p \in \beta (\alpha ):\;{\text {there is}}\;A \in p\;{\text {with}}\;|A| < \alpha \}$.