For a locally compact group G, the Chabauty space \({\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( G\right) \) of closed subgroups is a compact Hausdorff space. Although \({\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( H\right) \) is always a closed subspace of \({\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( G\right) \) for every closed subgroup H of G, it is not necessarily open. Understanding when \({\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( H\right) \) is open provides insight into the topological and algebraic structure of H and its interaction with G. In this paper, we investigate closed subgroups with open Chabauty subspaces and introduce the class \(\mathcal {T}\!\left( G\right) \) of such subgroups. Our first main theorem establishes a criterion relating open morphisms and Chabauty openness: for a continuous morphism \(\varphi :G\rightarrow H\) with G, H compact groups, the induced map \(\varphi _*:{\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( G\right) \rightarrow {\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( H\right) \) , defined by \(L\mapsto \varphi (L)\) , is open if and only if \(\varphi \) is open, provided that \(H_0\subseteq \varphi (G)\) . We also study isolated points of \({\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( G\right) \) when G is compact. We prove that a closed subgroup H is isolated in \({\mathcal {S\hspace{-0.5pt}U\hspace{-0.9pt}B}}\hspace{-0.6pt}\left( G\right) \) precisely when it is strongly finitely generated and belongs to \(\mathcal {T}\!\left( G\right) \) . This result generalizes previous work on discrete and profinite groups and yields a complete characterization in compact groups.
