Theorem 1

Let G be a locally compact group. The space $\Sigma$ of closed subgroups of G is compact. Moreover, one has the following properties:

(i) The set $\Sigma^0$ of unimodular closed subgroups of G is closed in $\Sigma$ (hence is compact).

(ii) If G is generated by a compact neighborhood of e, then the set $\Sigma_c^0$ of unimodular closed subgroups H of G such that the quotient space $G/H$ is compact, is open in $\Sigma^0$ (hence is locally compact).

(iii) For every relatively compact open neighborhood U of e in G, the set $D_U$ of discrete subgroups H of G such that $H \cap U = \{e\}$ is closed in $\Sigma^0$ (hence is compact).

It follows from Prop. 3 of No. 1 that $\Sigma$ is homeomorphic to $\Gamma_\varphi$, hence is compact by Prop. 2 of No. 1. Moreover, it was noted at the beginning of No. 2 that the set $\Gamma^0$ of measures $\alpha \in \Gamma$ such that $H_\alpha$ is unimodular is closed in $\Gamma$; since $\Gamma^0$ is stable under the homotheties with ratio $> 0$, the image $\Sigma^0$ of $\Gamma^0$ in $\Sigma$ is a closed subset of $\Sigma$, which proves (i).

Property (ii) will be a consequence of the following proposition: