Zero-dimensional proximities and zero-dimensional compactifications

Abstract

We introduce zero-dimensional proximities and show that the poset 〈 Z ( X ) , ⩽ 〉 of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset 〈 Π ( X ) , ⩽ 〉 of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that 〈 Π ( X ) , ⩽ 〉 is isomorphic to the poset 〈 B ( X ) , ⊆ 〉 of Boolean bases of X, and derive Dwinger's theorem that 〈 Z ( X ) , ⩽ 〉 is isomorphic to 〈 B ( X ) , ⊆ 〉 as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone–Čech compactification of X is a unique up to equivalence extremally disconnected compactification of X.