Exercise 15, § 3

a) Let $X$ be a uniform space, $\mathfrak{S}$ a set of parts of $X$, $\Gamma$ the group of homeomorphisms of $X$, $u_0$ an element of $\Gamma$. Suppose that for every $A \in \mathfrak{S}$ there exist a set $B \in \mathfrak{S}$ and an entourage $V$ of $X$ such that: $\alpha)$ $u_0^{-1}$ is uniformly continuous in $V(A)$; $\beta)$ the relations $u \in \Gamma$ and $(u_0(x), u(x)) \in V$ for all $x \in B$ imply $A \subset u(B)$. Under these conditions, prove that the mapping $u \mapsto u^{-1}$ (defined in $\Gamma$) is continuous at the point $u_0$ for the topology of $\mathfrak{S}$-convergence on $\Gamma$.

b) Prove that the condition $\beta)$ can be replaced by the following: $\beta')$ there exists a connected set $C \subset X$ such that $V(A) \subset C$ and that $V(C)$ is contained in the interior of $u_0(B)$ (prove that $\beta'$ implies $\beta$), by reasoning by contradiction.