Abstract

In this paper, we show that every compactification, which is a quasi-F space, of a space X is a Wallman compactification and that for any compactification K of the space X, the minimal quasi-F cover QFK of K is also a Wallman compactification of the inverse image <TEX>${\Phi}_K^{-1}(X)$</TEX> of the space X under the covering map <TEX>${\Phi}_K:QFK{\rightarrow}K$</TEX>. Using these, we show that for any space X, <TEX>${\beta}QFX=QF{\beta}{\upsilon}X$</TEX> and that a realcompact space X is a projective object in the category <TEX>$Rcomp_{\sharp}$</TEX> of all realcompact spaces and their <TEX>$z^{\sharp}$</TEX>-irreducible maps if and only if X is a quasi-F space.