On topological groups with remainder of character \(\kappa\) (Q2800910)

For a Tychonov topological space \(X\), a remainder is the subspace \(bX\setminus X\) of a Hausdorff compactification \(bX\) of \(X\). Moreover, denote by \(\chi(X)\) the character of \(X\).NEWLINENEWLINEIn this paper, the authors generalize results from \textit{A. V. Arhangel'skii} and \textit{J. van Mill} [Topol. Proc. 42, 157--163 (2013; Zbl 1285.54027)] to the case of an arbitrary infinite cardinal \(\kappa\). They prove that, if a topological group \(G\) is not locally compact and has a remainder such that \(\chi(bG\setminus G)=\kappa\), then \(\chi(G)\leq\kappa^+\) (so, \(|G|\leq 2^{\kappa^+}\)). This is the best possible estimation, since an example is given of a non locally compact topological group \(G\) with \(\chi(G)=\kappa^+\) and with a remainder such that \(\chi(bG\setminus G)=\kappa\).