For a locally compact group \(G\), denote by \(\mathcal{SUB}(G)\) the set of all closed subgroups of \(G\) endowed with the Chabauty topology, which is compact and Hausdorff. If \(H\) is a closed subgroup of \(G\), then the Chabauty topology of \(\mathcal{SUB}(H)\) coincides with the subspace topology of the Chabauty topology of \(\mathcal{SUB}(G)\) and \(\mathcal{SUB}(H)\) is a closed subspace of \(\mathcal{SUB}(G)\).
There are known cases when \(\mathcal{SUB}(H)\) is open in \(\mathcal{SUB}(G)\), but for example when \(G=\mathbb Z\) and \(H=\{0\}\) this does not occur. Let \(\mathcal T(G)\) be the family of all closed subgroups \(H\) of \(G\) such that \(\mathcal{SUB}(H)\) is open in \(\mathcal{SUB}(G)\). Clearly, \(G\in\mathcal T(G)\) and as an example \(\mathcal T(\mathbb Z)=\{\mathbb Z\}\).
This paper is focused on the family \(\mathcal T(G)\) when \(G\) is a compact group. For a closed subgroup \(H\) of \(G\), denote by \(N_G(H)\) the normalizer of \(H\) in \(G\), by \(G_0\) the connected component of \(1\) in \(G\) and let \(\mathcal M_G:=\{H\in\mathcal{SUB}(G):G_0\subseteq N_G(H)\}\). Moreover, let \begin{align*} &\mathcal{SUB}_n(G):=\{H\in\mathcal{SUB}(G): N\ \text{normal in}\ G\}\supseteq\mathcal N(G):=\{N\in\mathcal{SUB}_n(G):G/N\ \text{Lie group}\},\\ &\mathcal{SUB}_o(G):=\{H\in\mathcal{SUB}(G):H\ \text{open in}\ G\}. \end{align*}
Some of the main results of the present paper are the following:

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if \(H\in\mathcal{SUB}(G)\), then \(H\in\mathcal T(G)\) if and only of \(N_G(H)\) is open in \(G\) and \(N_G(H)/H\) is a Lie group;

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\(\mathcal T(G)=\{NH:N\in\mathcal{N}(G),H\in\mathcal M_G\}\) and the closure of \(\mathcal T(G)\) in \(\mathcal{SUB}(G)\) is \(\mathcal M_G\);

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if \(G\) is abelian, then \(\mathcal T(G)=\mathcal N(G)\) and \(\mathcal T(G)\) is dense in \(\mathcal{SUB}(G)\).

Moreover, some characterizations in the case of Lie groups are given:

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\(\mathcal T(G)=\mathcal M_G\) if and only if \(G\) is a Lie group;

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\(\mathcal T(G)=\mathcal{SUB}(G)\) if and only if \(G\) is a Lie group with \(G_0\) central;

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if \(G\) is a connected Lie group, then \(\mathcal T(G)=\mathcal{SUB}_n(G)\) and \(\mathcal T(G)\) is a closed in \(\mathcal{SUB}(G)\).

While in the totally disconnected case:

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\(G\) is totally disconnected if and only if \(\mathcal T(G)=\mathcal{SUB}_o(G)\) and \(\mathcal T(G)\) is dense in \(\mathcal{SUB}(G)\);

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\(G\) is topologically isomorphic to the group of \(p\)-adic integers \(\mathbb J_p\) for some prime \(p\) if and only if \(\mathcal T(G)=\mathcal{SUB}(G)\setminus \{\{1\}\}\) and \(\mathcal T(G)\) is an open dense subset of \(\mathcal{SUB}(G)\).