Proposition 10

Let $X$ be a uniform space and let $H$ be an equicontinuous set of homeomorphisms of $X$ onto itself. If $H$ and $H^{-1}$ are endowed with the topology of pointwise convergence in $X$, then the mapping $u \to u^{-1}$ of $H^{-1}$ onto $H$ is continuous.

It is enough to show that, for each $x_0 \in X$, the mapping $u \to u^{-1}(x_0)$ of $H^{-1}$ into $X$ is continuous at every point $u_0 \in H^{-1}$. Let $V$ be a symmetric entourage of $X$, and let $y_0 = u_0^{-1}(x_0)$. By hypothesis there is a symmetric entourage $U$ of $X$ such that the relation $(x, x_0) \in U$ implies $(u^{-1}(x), u^{-1}(x_0)) \in V$ for all $u \in H^{-1}$. Take an element $u \in H^{-1}$ which is $W(\{y_0\}, U)$-close to $u_0$; then we have $(u(y_0), u_0(y_0)) \in U$, i.e. $(u(y_0), x_0) \in U$. It follows that $(y_0, u^{-1}(x_0)) \in V$, i.e. $(u_0^{-1}(x_0), u^{-1}(x_0)) \in V$; this completes the proof.