A theorem on remainders of topological groups
Recall that a \textit{remainder} of a Tychonoff topological space \(X\) is the subspace \(bX\setminus X\) of a Hausdorff compactification \(bX\) of \(X\). Moreover, \(X\) is \textit{perfect} if every closed subset of \(X\) is a \(G_\delta\)-set of \(X\). By a result in [\textit{A. V. Arhangel'skii}, Fundam. Math. 203, No. 2, 165--178 (2009; Zbl 1182.54043)], for a remainder \(Y\) of a topological group \(G\), if \(Y\) is perfect then \(Y\) is first-countable. The main theorem of the present paper extends this result in case \(G\) is non-locally compact, by showing that \(Y\) perfect implies that \(G\) is separable and metrizable, \(Y\) is hereditarily Lindelöf and first-countable; moreover, \(Y\) is a \(p\)-space with countable \(\pi\)-base and every compact subspace of \(Y\) has a countable base of open neighborhoods in \(Y\). An example shows that \(Y\) needs not be necessarily metrizable. Moreover, it is known that there exists a non-locally compact topological group that is not metrizable but admits a first-countable remainder. Consequences of the main theorem are that a remainder of a topological group is perfect precisely when it is hereditarily Lindelöf (this answers Problem 3.9 from [loc. cit.]), and that a non-locally compact topological group is separable and metrizable if and only if it admits a perfect remainder.
scientific article; zbMATH DE number 6695391
scientific article