The set \({ \mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\) of closed subgroups of a locally compact group \(G\) has a natural topology, called the Chabauty topology, which makes it a Hausdorff compact space. This topology has been defined by C. Chabauty [Bull. Soc. Math. Fr. 78, 143–151 (1950; Zbl 0039.04101)]. In the paper under review the authors study in detail the continuity of the following functions  \(G\to {\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\) defined by \[ \begin{aligned} & \mu_G(g) = \overline{\langle g\rangle},\text{ the closed subgroup generated by }g;\\ &\mathrm{lev}_G(g) = \left\{x\in G \mid \{g^kxg^{-k}\}_{k\in \mathbb{Z}}\text{ is precompact}\right\},\text{ the \textit{Levi} subgroup of }g;\text{ and }\\ &\mathrm{par}_G(g) = \left\{x\in G \mid \{g^kxg^{-k}\}_{k\in \mathbb{N}} \text{ is precompact}\right\},\text{ the \textit{parabolic} subgroup of }g. \end{aligned} \] Among the authors’ several results are these:
(Theorem 1): For a totally disconnected, locally compact group \(G\), the functions \(\mu_G\), \(\mathrm{lev}_G\) and \(\mathrm{par}_G\) are continuous from \(G\) to \({\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\).
(Theorem 4): The function \(\mu_G: G \to {\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\) is continuous on \(G\) if and only if the locally compact group \(G\) is totally disconnected.
Some further open questions about the continuity of certain functions \(G\to {\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\) are mentioned at the end of the paper.