Exercise 7, § 2

Let X be a topological space, Y a complete Hausdorff uniform space, and let $\varphi$ be a uniformly continuous homeomorphism of Y onto an open subset $\varphi(Y)$ of a Hausdorff uniform space $Y'$. Let H be an equicontinuous subset of $\mathscr{C}(X; Y)$, and let H' be the set of mappings $\varphi \circ u$, where $u \in H$. If $v$ lies in the closure of H' in $\mathscr{F}_s(X; Y')$, show that $\overline{\nu}^1(\varphi(Y))$ is both open and closed in X. In particular, if X is connected, $\nu(X)$ is contained in $\varphi(Y)$ or in a connected component of $C_{\varphi}(Y)$. (Observe that $\nu$ is continuous, and use Exercise 6.)