A locally compact group $G$ is called compactly ruled if it is a directed union of compact open subgroups. We denote by $\mathcal{SUB}(G)$ the space of closed subgroups of $G$ equipped with the Chabauty topology. In this paper, we show that the subspace $\mathcal{SUB}_{\rm co} (G)$ of $\mathcal{SUB}(G)$ consisting of compact open subgroups is dense in $\mathcal{SUB}(G)$ if and only if $G$ is totally disconnected compactly ruled.

Given a locally compact group $G$, we denote by $\mathcal{SUB}(G)$ the set of closed subgroups of $G$ equipped with the Chabauty topology, which is a compact Hausdorff. For a closed subgroup $H$ of $G$, the Chabauty topology on $\mathcal{SUB}(H)$ is equivalent to the subspace topology on $\{L \in \mathcal{SUB}(G) \mid L \subseteq H\}$ inherited from $\mathcal{SUB}(G)$, and so $\mathcal{SUB}(H)$ becomes a closed subset of $\mathcal{SUB}(G)$. In some cases, the subspace $\mathcal{SUB}(H)$ is open in $\mathcal{SUB}(G)$, but it can also fail to be open as the example of the trivial subgroup $\{0\}$ of the additive group $\mathbb{Z}$ shows. In this paper, we are interested to determine the subset $\mathcal{T} (G)$ of closed subgroups $H$ of a compact group $G$ such that $\mathcal{SUB}(H)$ is open in $\mathcal{SUB}(G)$. An explicit description of $\mathcal{T} (G)$ is given. The paper also contains some topological properties of $\mathcal{T} (G)$. As an application, we show that if $G$ is a compact group and if the lattice $\mathcal{SUB}(G)$ is isomorphic to the lattice $\mathcal{SUB}(\mathbb{Z}_p)$ of the group $\mathbb{Z}_p$ of $p$-adicintegers, then $G$ is topologically isomorphic to $\mathbb{Z}_l$, for some prime $l$.