Corollary

Let $G$ be a locally compact abelian group, generated by a compact neighborhood of $e$ and satisfying the condition (L) of No. 4.

Let $D_c$ be the set of discrete subgroups $H$ of $G$ such that $G/H$ is compact, and, for every $H \in D_c$, let $v(H)$ be the total mass $\mu_\alpha(G/H)$, where $\mu_\alpha$ is the quotient measure of $\mu$ by the normalized Haar measure $\alpha$ of $H$. For a subset $A$ of the space $D_c$ to be relatively compact in $D_c$, it is necessary and sufficient that it satisfy the following two conditions:

(i) There exists an open neighborhood $U$ of $e$ in $G$ such that $H \cap U = \{e\}$ for all $H \in A$.

(ii) There exists a constant $k$ such that $v(H) \leq k$ for all $H \in A$.

Taking into account Prop. 9, this follows at once from the fact that $D_c$ is the image of $N_c$ under the canonical bijection of $N$ onto $D$, and the fact that, under the hypotheses made, this bijection is a homeomorphism (No. 4, Prop. 8).