Wallman-type compactifications on 0-dimensional spaces

Abstract

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