  Let $\mathcal{C}(G)$ denote the Chabauty space of closed subgroups of the
locally compact group $G$. In this paper, we first prove that $\mathcal{C}
(\mathbb{Q}_p^\times)$ is a proper compactification of $\mathbb{N}$, identified
with the set $N$ of open subgroups with finite index. Then we identify the
space $\mathcal{C}(\mathbb{Q}_p^\times) \smallsetminus N$ up to homeomorphism:
e.g. for $p=2$, it is the Cantor space on which 2 copies of
$\overline{\mathbb{N}}$ (the 1-point compactification of $\mathbb{N}$) are
glued.
