Exercise 12, § 2

Let $X$ be a topological space, $Y$ a uniform space, $H$ an equicontinuous subset of $C(X; Y)$ and $V$ a symmetric entourage of $Y$. Given a point $x \in X$, show that the set of points $x' \in X$ for which there is an integer $n$ (depending on $x'$) such that $H(x') \subset \overset{n}{V}(H(x))$ is both open and closed in $X$. Deduce that, if $K$ is any compact connected subset of $X$ and if $x_0$ is any point of $K$, there is an integer $n > 0$ such that $H(K) \subset \overset{n}{V}(H(x_0))$.