Wallman covers and quasi-\(F\) covers (Q2870893)

Quasi-\(F\) spaces were introduced in [\textit{F. Dashiell} et al., Can. J. Math. 32, 657--685 (1980; Zbl 0462.54009)] as Tychonoff spaces in which every dense cozero-set is \(C^*\)-embedded. A quasi-\(F\) cover of a Tychonoff space \(X\) is defined as a pair \((Y,f)\), where \(Y\) is a quasi-\(F\) space and \(f\) is an irreducible perfect function that maps \(Y\) continuously onto \(X\). It is a result of \textit{J. Vermeer}, [Topology Proc. 9, No. 1, 173--189 (1984; Zbl 0561.54029)] that every Tychonoff space \(X\) admits a (unique) minimal quasi-\(F\) cover \((QFX,\Phi)\), where ``minimal'' means that, for every cover \((Z,g)\) of \(X\), there is \(h: Z \rightarrow QFX\) such that \((Z,h)\) is a cover of \(QFX\) satisfying \(\Phi \circ h = g\).NEWLINENEWLINEIn the present paper, the authors consider the Wallman sublattice \(\mathscr{A}_X=\{cl(int(\Phi(A))):A\in Z(QFX)^{\#}\}\) of the lattice \(\mathscr{R}(X)\) of regular closed subsets of a Tychonoff space \(X\), where \(Z(QFX)^{\#}=\{cl(int(S)):S\) is a zero-set of \(QFX\}\). As shown in [\textit{M. Henriksen} et al., Trans. Am. Math. Soc. 303, 779--803 (1987; Zbl 0653.54025)], the set \(\mathscr{L}(\mathscr{A}_X)\) of all of the ultrafilters on \(\mathscr{A}_X\) is naturally endowed with a compact Hausdorff topology that has \(\{A^*:A\in\mathscr{A}_X\}\) as a closed base, where \(A^*=\{\alpha\in\mathscr{L}(\mathscr{A}_X):A\in\alpha\}\) for \(A\in\mathscr{A}_X\).NEWLINENEWLINEIn this paper, the authors show that \(QFX\) is homeomorphic to the subspace \(X_q=\{\alpha\in\mathscr{L}(\mathscr{A}_X):\alpha\) is non-principal\(\}\subseteq\mathscr{L}(\mathscr{A}_X)\), and that \(\beta QFX\) is homeomorphic to \(\mathscr{L}(\mathscr{A}_X)\).