Wallman-type compactifications on $0$-dimensional spaces

Abstract:

Let $E$ be Hausdorff $0$-dimensional, $\mathcal {D}$ the discrete space $\{0, 1\}$, and $\mathcal {N}$ the discrete space of all nonnegative integers. Every $E$-completely regular space $X$ has a clopen normal base $\mathcal {F}$ with $X\backslash F \in \mathcal {F}$ for each $F \in \mathcal {F}$. The Wallman compactification $\omega (\mathcal {F})$ is $\mathcal {D}$-compact. Moreover, if an $E$-completely regular space $X$ has a countably productive clopen normal base $\mathcal {F}$ with $X\backslash F \in \mathcal {F}$ for each $F \in \mathcal {F}$, then the Wallman space $\eta (\mathcal {F})$ is $\mathcal {N}$-compact. Hence, if $X$ has such an $\mathcal {F}$, and is an $\mathcal {F}$-realcompact space, then $X$ is $\mathcal {N}$-compact.