Exercise 6, § 3

a) Let $X$ be a topological space, $Y$ a metric space, $d$ the metric on $Y$, $n$ an integer $> 0$ and $S$ a subset of the space $X^n \times Y^n \times \mathbf{R}_+^*$. For each point $z = ((x_i), (y_i), r) \in S$, let $U_z$ be the open subset of $\mathscr{C}_s(X; Y)$ consisting of continuous mappings $f$ of $X$ into $Y$ such that $d(f(x_i), y_i) < r$ for $1 \leq i \leq n$. Show that, if D is a dense subset of S, we have $\bigcup_{z \in S} U_z = \bigcup_{z \in D} U_z$.

b) Deduce from a) that if the topologies of X and Y have countable bases, then every subspace of $\mathscr{C}_s(X; Y)$ is a Lindelöf space (Chapter I, § 9, Exercise 15) and is therefore paracompact (Chapter IX, § 4, Exercise 23).