Zero-dimensional proximities and zero-dimensional compactifications
It is shown that the poset of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space \(X\) is isomorphic to the poset of zero-dimensional proximities on \(X\) that induce the topology on \(X\). That poset is also isomorphic to the poset of Boolean bases of \(X\). The author also states that for a regular extremally disconnected space \(X\), the Stone-Čech compactification of \(X\) is a unique (up to equivalence) extremally disconnected compactification of \(X\). We note that related investigations about transitive totally bounded quasi-uniformities were conducted in asymmetric topology, see e.g., the article of \textit{A. Losonczi} [Quaest. Math. 23, No.~1, 67--75 (2000; Zbl 0958.54033)] where lattice-bases were studied or that of \textit{H. P. A. Künzi} and \textit{G. C. L. Brümmer} [Math. Proc. Camb. Philos. Soc. 101, 237--247 (1986; Zbl 0618.54024)] where those compact \(T_2\)-ordered spaces whose determining (totally bounded) quasi-uniformity is transitive were characterized.
scientific article; zbMATH DE number 5549312