Exercise 8, § 2

Let H be an equicontinuous set of mappings of a topological space X into $\mathbf{R}$. a) Show that the set of upper (resp. lower) envelopes of finite subsets of H is equicontinuous. b) Let $v$ be a mapping of X into $\overline{\mathbf{R}}$ which lies in the closure of H with respect to the topology of pointwise convergence. Show that $v$ is continuous on $X$ and that the sets $\bar{v}^1(+\infty)$ and $\bar{v}^1(-\infty)$ are both open and closed in $X$ (use Exercise 7).

c) Deduce from a) and b) that the upper (resp. lower) envelope $w$ (resp. $v$) of $H$ is continuous on $X$ and that $\bar{w}^1(+\infty)$ [resp. $\bar{v}^1(-\infty)$] is both open and closed in $X$.

d) Suppose that $X$ is connected and that there is a point $x_0 \in X$ such that $H(x_0)$ is bounded above in $\mathbf{R}$. Show that, for every compact subset $K$ of $X$, the set of restrictions to $K$ of functions of $H$ is uniformly bounded above in $K$ [use c)].