Wallman-Type Compactifications and Products

Abstract

Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted $\omega (Z)$, is topologically Y. It is not known if every compactification is Wallman-type. For ${Z_\alpha }$ a normal base for the closed sets of ${X_\alpha }$ for each a belonging to an index set $\Delta$ it is shown that the Tychonoff product space ${\prod _{\alpha \in \Delta }}\omega ({Z_\alpha })$ is a Wallman compactification of ${\prod _{\alpha \in \Delta }}{X_\alpha }$. Also for $X \subset T \subset \omega (Z)$ with Z a normal base for the closed sets of X, a proof that $\omega (Z)$ is a Wallman-type compactification of T is indicated.