Corollary

Let $G$ be a group of homeomorphisms of a locally compact space $X$. If the closure $\overline{G}$ of $G$ in $C_c(X; X)$ is compact, then $\overline{G}$ is a group of homeomorphisms of $X$, and the topology of compact convergence is compatible with the group structure of $\overline{G}$, which is therefore a compact topological group.

A group of homeomorphisms of a locally compact space $X$ which is locally compact but not compact with respect to the topology of compact convergence is locally closed in $C_c(X; X)$ by virtue of Chapter I, § 9, no. 7, Proposition 12, but is not necessarily closed.

For example, in the ring $\mathscr{L}(\mathbf{R}^n)$ of endomorphisms of $\mathbf{R}^n$, identified with the ring $M_n(\mathbf{R})$ of square $n \times n$ matrices over $\mathbf{R}$ and endowed with the topology of compact convergence, the group $GL(n, \mathbf{R})$, identified with the group of non-singular matrices, is locally compact but dense (Chapter VI, § 1, no. 6, Proposition 6).