  The Chabauty space of a topological group is the set of its closed subgroups,
endowed with a natural topology. As soon as $n>2$, the Chabauty space of $R^n$
has a rather intricate topology and is not a manifold. By an investigation of
its local structure, we fit it into a wider, but too wild, class of topological
spaces (namely Goresky-MacPherson stratified spaces). Thanks to a localization
theorem, this local study also leads to the main result of this article: the
Chabauty space of $R^n$ is simply connected for all $n$. Last, we give an
alternative proof of the Hubbard-Pourezza Theorem, which describes the Chabauty
space of $R^2$.
