Theorem 4

Let $G$ be a subgroup of the group $\Gamma$ of all homeomorphisms of a locally compact space $X$. Suppose that, in the space $C_c(X; X)$, there is a neighbourhood $V$ of the identity mapping $e$ such that $V \cap G = H$ is symmetric in $G$ and relatively compact in $C_c(X; X)$. Then the closure $\overline{G}$ of $G$ in $\Gamma$ with respect to the topology $\mathscr{T}_\beta$ is a locally compact group with respect to the topology induced by $\mathscr{T}_\beta$; this topology induced on $\overline{G}$ is the same as the topology of compact convergence, and the closure $\overline{H}$ of $H$ in $C_c(X; X)$ is a neighbourhood of $e$ in $\overline{G}$ with respect to this topology.

Let us show first that $\overline{H}$ is contained in $\Gamma$ and that the topology induced on $\overline{H}$ by $\mathscr{T}_\beta$ is the same as the topology of compact convergence. Let $u_0 \in \overline{H}$; $u_0$ is therefore the limit, in $C_c(X; X)$, of an ultrafilter $\Phi$ on $H$. Since $\Phi^{-1}$ (the image of $\Phi$ under $u \to u^{-1}$) is an ultrafilter base on $H \subset \overline{H}$, it converges in the compact subspace $\overline{H}$ of $C_c(X; X)$ to an element $v_0$. The mapping $(u, v) \to uv$ converges to $u_0 v_0$ with respect to $\Phi \times \Phi^{-1}$ (no. 4, Proposition 9); a fortiori, $u \to uu^{-1} = e$ converges to $u_0 v_0$ with respect to $\Phi$, hence $u_0 v_0 = e$ since $C_c(X; X)$ is Hausdorff. Similarly $v_0 u_0 = e$; hence $u_0$ is a homeomorphism of $X$, i.e., $u_0 \in \Gamma$. Thus $\overline{H}$ is contained in $\Gamma$. Moreover, this argument shows that $\overline{H}^{-1} = \overline{H}$ and that, for every ultrafilter $\Phi$ on $\overline{H}$ which converges to $u_0$, $\Phi^{-1}$ converges in $C_c(X; X)$ to $u_0^{-1}$; hence the mapping $u \to u^{-1}$ of $\overline{H}$ into $C_c(X; X)$ is continuous when $\overline{H}$ carries the topology of compact convergence (Chapter I, § 7, no. 4, Proposition 9, Corollary 1). Proposition 12 then shows that, on $\overline{H}$, the topology of compact convergence is the same as the topology induced by $\mathscr{T}_\beta$.

Furthermore, since the topology $\mathscr{T}_\beta$ on $\Gamma$ is finer than the topology of compact convergence, $\overline{H}$ is also the closure of $H$ with respect to $\mathscr{T}_\beta$. But $H$ is a neighbourhood of $e$ in $G$ with respect to the topology of compact convergence, and a fortiori with respect to the topology induced by $\mathscr{T}_\beta$; it follows (Chapter I, § 3, no. 1, Proposition 2) that $\overline{H}$ is a neighbourhood of $e$ in $\overline{G}$ with respect to the topology induced by $\mathscr{T}_\beta$, and hence $\overline{G}$ is locally compact in this topology. If $W$ is the interior of $V$ with respect to the topology of compact convergence, then $W \cap \Gamma$ is open in $\mathscr{T}_\beta$, hence $W \cap \overline{G}$ is contained in the closure of $H = V \cap G$ with respect to $\mathscr{T}_\beta$ (Chapter I, § 1, no. 6, Proposition 5); this shows that $\overline{H}$ is also a neighbourhood of $e$ in $\overline{G}$ with respect to the topology of compact convergence. Finally, for each $u_0 \in \Gamma$, the bijections $v \to u_0 \circ v$ and $v \to u_0^{-1} \circ v$ of $C_c(X; X)$ onto itself are continuous (no. 4, Proposition 9), and hence, if $u_0 \in \overline{G}$, $u_0 \overline{H}$ is a neighbourhood of $u_0$ in $\overline{G}$ with respect to the topology of compact convergence. This completes the proof.