#### Vol. 14, No. 1, 2021

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

### Antoine Bourquin and Alain Valette

Vol. 14 (2021), No. 1, 89–102
##### Abstract

Let $\mathsc{𝒞}\left(G\right)$ denote the Chabauty space of closed subgroups of the locally compact group $G$. We first prove that $\mathsc{𝒞}\left({ℚ}_{p}^{×}\right)$ is a proper compactification of $ℕ$, identified with the set $N$ of open subgroups with finite index. Then we identify the space $\mathsc{𝒞}\left({ℚ}_{p}^{×}\right)\N$ 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 $ℕ$) are glued.

##### Keywords
Chabauty space, locally compact group, $p$-adic group, topological space, proper compactification
##### Mathematical Subject Classification 2010
Primary: 22B05, 54H11
##### Milestones
Received: 4 November 2019
Revised: 11 July 2020
Accepted: 15 September 2020
Published: 4 March 2021

Communicated by Kenneth S. Berenhaut
##### Authors
 Antoine Bourquin Institut de Mathématiques Université de Neuchâtel Neuchâtel Switzerland Alain Valette Faculté des Sciences Institut de Mathématiques Université de Neuchâtel Neuchâtel Switzerland