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Let  G  be a locally compact group. We denote by ℬ(ℝ ×  G ) the hyperspace of closed subgroups of ℝ ×  G  endowed with the Chabauty topology. In this paper, some general results about the connectedness of the hyperspace ℬ(ℝ ×  G ) are established. The main result proved here is that if  G  is a prosolvable group, then ℬ(ℝ ×  G ) is connected. We also prove a partial converse to the result just mentioned.

On the connectedness of the Chabauty space of a locally compact prosolvable group