Local properties on the remainders of the topological groups (Q655541)

When does a Tychonoff space \(X\) or a topological group \(G\) have a Hausdorff compactification with remainder belonging to a given class of spaces? Many topologists studied the question, in particular Henriksen and Isbell obtained one of the famous results in this study such as follows: A space \(X\) is of countable type if and only if the remainder in any (in some) compactification of \(X\) is Lindelöf. For a non-locally compact topological group \(G\), Arhangel'skiĭ\ showed that if the remainder has a \(G_{\delta}\)-diagonal or a point-countable base, then both the remainder and \(G\) are separable and metrizable, a result that was improved by \textit{C. Liu} [Topology Appl. 156, No. 5, 849--854 (2009; Zbl 1162.54007)]. Furthermore, in this paper the author improves the above results for a topological group. Let \(G\) be a non-locally compact topological group and \(bG\) a compactification of \(G\). He proves that both \(G\) and \(bG\setminus G\) are separable and metrizable if one of the following holds: (i) \(bG\setminus G\) is locally a \(k\)-space with a point-countable \(k\)-network and the \(\pi\)-character of \(bG\setminus G\) is countable; (ii) \(bG\setminus G\) has locally a \(\delta\theta\)-base; (iii) \(bG\setminus G\) has locally a quasi-\(G_{\delta}\)-diagonal; (iv) \(bG\setminus G\) is Ohio complete with a base of countable order. The last result is a partial answer to a question posed by Liu [loc.cit.].