Theorem 2

— Let $G$ be a locally compact group satisfying the following condition: (L) There exists a neighborhood of $e$ in $G$ that contains no finite subgroup of $G$ not reduced to $e$. The following properties then hold: (i) The set $D$ of discrete subgroups of $G$ is locally closed in $\Sigma$ (which is equivalent to saying that it is locally compact). (ii) For a closed subset $A$ of $D$ to be compact, it is necessary and sufficient that there exist a neighborhood $U$ of $e$ in $G$ such that $H \cap U = \{e\}$ for every subgroup $H \in A$. (iii) If in addition $G$ is generated by a compact neighborhood of $e$, then the set $D_c$ of discrete subgroups $H$ of $G$ such that $G/H$ is compact is locally closed in $\Sigma$ (hence is locally compact).

We have $D_c = D \cap \Sigma_c^0$, therefore (iii) is a consequence of (i), and Th. 1 (ii) of No. 3.

With the notations of No. 3, Prop. 7, it suffices, for proving (i) and (ii), to prove that: