Weakly submaximal spaces and compactifications (Q1690543)

A space \(X\) is said to be weakly submaximal if each finite subset of \(X\) is locally closed. In the paper under review, the authors provide information about the space \(X\) when a compactification \(K(X)\) is weakly submaximal (resp., a \(T_D\)-space). If \(\widetilde{X}\) denotes the one-point compacification of \(X\), then the authors show the following properties: 1. \(\widetilde{X}\) is a \(T_D\)-space if and only if so is \(X\). 2. \(\widetilde{X}\) is weakly submaximal if and only if each finite subset of \(X\) is the intersection of a closed set of \(X\) and a co-compact open set of \(X\). Moreover, the authors provide characterizations of spaces with weakly submaximal (resp., \(T_D\)) Herrlich compactification. The proofs are clear and elementary.