On the continuity of the centralizer map of a locally compact group (Q2810923)

Let \(G\) be a locally compact group. The set \(\mathrm{SUB}(G)\) of all closed subgroups of \(G\) is endowed with the Chabauty topology (defined by \textit{C. Chabauty} [Bull. Soc. Math. Fr. 78, 143--151 (1950; Zbl 0039.04101)]). The paper under review studies the continuity of the centralizer map \(\mathrm{Cent}_G: G\to \mathrm{SUB}(G)\) defined by \(\mathrm{Cent}_G(t)=\{s\in G: ts=st\}\) for each \(t\in G\) (this question was initially addressed by \textit{K. H. Hofmann} and \textit{G. A. Willis} [J. Lie Theory 25, No. 1, 1--7 (2015; Zbl 1317.22004)]). The authors show that the map \(\mathrm{Cent}_G\) is upper semicontinuous for each locally compact group. If the locally compact group \(G\) has an open abelian subgroup, then the map \(\mathrm{Cent}_G\) is continuous on \(G\) if and only if it is continuous at the identity \(e\) of \(G\). Further, if for a locally compact group \(G\) the map \(\mathrm{Cent}_G\) is continuous at \(e\), then the component of \(e\) is central.