Exercise 6, § 5

Let G be a locally compact group that does not satisfy the condition (L) of No. 4. a) Show (with the notations of No. 4) that the canonical bijection of N onto D is not a homeomorphism. (For every neighborhood W of e in G, let A(W) be the set of finite subgroups $\neq \{e\}$ contained in W; show that the A(W) form the base of a filter that converges in D to the subgroup $\{e\}$, but that if $\alpha_H$ denotes the normalized Haar measure of H, the measures $\alpha_H$ do not converge to $\varepsilon_e$.) b) If in addition G is metrizable, show that there exists a compact subset B of the space D of discrete subgroups of G, that is not contained in any of the sets $D_U$ defined in Th. 1 of No. 3.