Exercise 13, § 2

Let $X$ be a topological space, let $Y$ be a locally compact uniform space and let $H$ be an equicontinuous subset of $C(X; Y)$.

a) Let $A$ be the set of all $x \in X$ such that $H(x)$ is relatively compact in $Y$. Show that $A$ is open in $X$ (cf. Chapter I, § 9, no. 7, Proposition 10 and Chapter II, § 4, no. 3, Proposition 4).

b) Suppose in addition that $Y$ is complete with respect to its uniformity; then $A$ is also closed in $X$ [observe that if $x_0 \in \overline{A}$, then $H(x_0)$ is relatively compact in $Y$ by considering an ultrafilter on $H(x_0)$ as the image of an ultrafilter on $H$]. In this case, if $X$ is connected, then $H$ is relatively compact in $C_c(X; Y)$ if and only if, for one point $x_0 \in X$, the set $H(x_0)$ is relatively compact in $Y$.

c) Take $X$ to be the compact interval $[0, 1]$ of $\mathbf{R}$, and take $Y$ to be the interval $[0, 1[$ endowed with the uniformity induced by that of $\mathbf{R}$. Give an example of an equicontinuous subset $H$ of $C(X; Y)$ such that the set $A$ defined in a) is the interval $[0, 1]$.

d) Suppose that $X = Y$ and that $H$ consists of homeomorphisms of $X$ onto itself. Show that if $H$ is uniformly equicontinuous, then the set $A$ defined in a) is both open and closed in $X$.