An arbitrary Hausdorff compactification of a Tychonoff space \(X\) obtained from a \(C_D^*\)-base by a modified Wallman method (Q929972)

The authors describe a modified Wallman method which they use to obtain any Hausdorff compactification \((Z,h)\) of a Tikhonov space \(X\). For this purpose they use special bases (for \(X\)) called \({\mathbb C}_{\mathbb D}^*\)-bases. Let \({\mathbb D} = \{g\circ h: g \in C(Z)\}\) and let \(\mathbb C\) be the collection of all nonempty sets of the form \(f^{\gets}([a,b])\), \(a,b\in \mathbb R\), \(a< b\). The family of all nonempty finite intersections of elements of \(\mathbb C\) is a base for closed sets in \(X\) and it is called a \({\mathbb C}_{\mathbb D}^*\)-base.