The Chabauty space of $\mathbb{Q}_p^\times$

Abstract

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