The Chabauty space of \(\mathbb{Q}_p^\times\) (Q2037250)

For a locally compact space \(X\), \textit{C. Chabauty} [Bull. Soc. Math. Fr. 78, 143--151 (1950; Zbl 0039.04101)] introduced a topology on the set \(\mathcal F(X)\) of all closed subsets of \(X\), turning it into a compact space. When \(X=G\) is a locally compact group \(G\), the family \(\mathcal C(G)\) of all closed subgroups of \(G\) is a closed (and so compact) subspace of \(\mathcal F(G)\), and \(\mathcal C(G)\) is called the Chabauty space of \(G\). This paper studies the Chabauty space of the multiplicative group \(\mathbb Q_p^*\) of the field of \(p\)-adic numbers \(\mathbb Q_p\). By results in [\textit{Y. Cornulier}, Algebr. Geom. Topol. 11, No. 4, 2007--2035 (2011; Zbl 1221.22008)] \(\mathcal C(\mathbb Q_p^*)\) is totally disconnected and uncountable; further properties are given in the present paper. Recalling that a compact metric space \(Y\) is a proper compactification of \(\mathbb N\) if \(Y\) contains a countable, open, dense and discrete subset \(N\), the main theorem states that \(\mathcal C(\mathbb Q_p^*)\) is a proper compactification of \(\mathbb N\), identified with the subset \(N\) of \(\mathcal C(\mathbb Q_p^*)\) of all open subgroups of \(G\) with finite index. Moreover, the space \(\mathcal C(\mathbb Q_p^*)\setminus N\) is described in details. This result gives also a contribution to a more general question by \textit{Y. Cornulier} [loc. cit.].