  Given a computably locally compact Polish space $M$, we show that its 1-point
compactification $M^*$ is computably compact. Then, for a computably locally
compact group $G$, we show that the Chabauty space $\mathcal S(G)$ of closed
subgroups of $G$ has a canonical effectively-closed (i.e., $\Pi^0_1$)
presentation as a subspace of the hyperspace $\mathcal K(G^*)$ of closed sets
of $G^*$. We construct a computable discrete abelian group $H$ such that
$\mathcal S(H)$ is not computably closed in $\mathcal K(H^*)$; in fact, the
only computable points of $\mathcal S(H)$ are the trivial group and $H$ itself,
while $\mathcal S(H)$ is uncountable. In the case that a computably locally
compact group $G$ is also totally disconnected, we provide a further
algorithmic characterization of $\mathcal S(G)$ in terms of the countable meet
groupoid of $G$ introduced recently by the authors (arXiv: 2204.09878). We
apply our results and techniques to show that the index set of the computable
locally compact abelian groups that contain a closed subgroup isomorphic to
$(\mathbb{R},+)$ is arithmetical.
