Exercise 15, § 3

a) Let $X$ be a uniform space, let $\mathscr{S}$ be a set of subsets of $X$, let $\Gamma$ be the group of all homeomorphisms of $X$, and let $u_0 \in \Gamma$. Suppose that for each $A \in \mathscr{S}$ there exist $B \in \mathscr{S}$ and an entourage $V$ of $X$ such that: $\alpha)$ $u_0^{-1}$ is uniformly continuous on $V(A)$; $\beta)$ the relations $u \in \Gamma$ and $(u(x_0), u(x)) \in V$ for all $x \in B$ together imply $A \subset u(B)$. Under these conditions, show that $u \to u^{-1}$ (defined on $\Gamma$) is continuous at $u_0$ with respect to the topology of $\mathscr{S}$-convergence on $\Gamma$.

b) Show that the condition $\beta)$ may be replaced by the following: $\beta')$ there exists a connected set $C \subset X$ such that $V(A) \subset C$ and $V(C)$ is contained in the interior of $u_0(B)$. [Show that $\beta') \Longrightarrow \beta)$ by reductio ad absurdum.]