Proposition 11

Let $X$ be a compact space and let $\Gamma$ be the group of all homeomorphisms of $X$ onto itself. Then the topology of uniform convergence in $X$ is compatible with the group structure of $\Gamma$.

We know already (no. 4, Proposition 9) that the mapping $(u, v) \to v \circ u$ of $\Gamma \times \Gamma$ into $\Gamma$ is continuous with respect to this topology; thus we have to show that $u \to u^{-1}$ is continuous at every point $u_0$ of $\Gamma$. Since $u_0^{-1}$ is uniformly continuous on $X$, given any symmetric entourage $V$ of $X$ there exists an entourage $W$ of $X$ such that the relation $(x, x') \in W$ implies $(u_0^{-1}(x), u_0^{-1}(x')) \in V$. Hence, if $u \in \Gamma$ is such that $(u_0(x), u(x)) \in W$ for all $x \in X$, it follows that $(x, u_0^{-1}(u(x))) \in V$ for all $x \in X$, and therefore (as $u$ is bijective) $(u^{-1}(x), u_0^{-1}(x)) \in V$ for all $x \in X$. This completes the proof.

Now let $X$ be a locally compact space and let $\Gamma$ be the group of all homeomorphisms of $X$ onto itself. The topology of compact convergence in $X$ is not necessarily compatible with the group structure of $\Gamma$ (Exercise 17). Let $X'$ denote the compact space obtained by adjoining a point at infinity $\omega$ to $X$. Every homeomorphism $u$ of $X$ onto itself extends uniquely to a homeomorphism $u'$ of $X'$ onto itself such that $u'(\omega) = \omega$ (Chapter I, § 10, no. 3, Corollary to Proposition 7), so that $\Gamma$ can be identified with the subgroup of the group $\Gamma'$ of all homeomorphisms of $X'$ onto itself, consisting of all homeomorphisms which leave $\omega$ fixed. The topology induced on $\Gamma$ by that of $C_u(X'; X')$ is therefore compatible with the group structure of $\Gamma'$ (Proposition 11), and $\Gamma$ is closed in $\Gamma'$ [with respect to the topology induced by that of $C_u(X'; X')$] because it is defined by the equation $u(\omega) = \omega$ (§ 1, no. 2, Remark 6). We denote by $\mathscr{T}_\beta$ the group topology thus defined on $\Gamma$; it is finer than the topology of compact convergence and can also (by virtue of § 1, no. 6, Proposition 6) be defined as the topology of uniform convergence on $X$, when $X$ is endowed with the uniformity induced by the unique uniformity of $X'$.

The topology $\mathscr{T}_\beta$ can be characterized as follows: