Exercise 11, § 3

Let $X, Y$ be two non-empty topological spaces whose product $X \times Y$ is normal. Let $\mathscr{T}$ be a topology on the set $C(X; \mathbf{R})$ such that, for every continuous real-valued function $f$ on $X \times Y$, the mapping $y \to f(., y)$ of $Y$ into $C(X; \mathbf{R})$ is continuous.

a) Let $y_0$ be a limit of a sequence $(z_n)$ of points of $Y$ all distinct from $y_0$, let $A$ be a countably infinite closed subset of $X$, all of whose points are isolated, and let $I$ be a bounded open interval in $\mathbf{R}$. Show that, with respect to the topology $\mathscr{T}$, the set $T(A, I)$ (Exercise 10) has no interior point. [If $u_0 \in T(A, I)$, construct a continuous mapping $f : X \times Y \to \mathbf{R}$ such that $f(., y_0) = u_0$ and that $f(., z_n) \notin T(A, I)$ for all $z_n \neq y_0$.]

b) Hence show that if, in addition, the mapping $(u, x) \to u(x)$ of $C(X; \mathbf{R}) \times X$ into $\mathbf{R}$ is continuous with respect to the topology $\mathscr{T}$, if $X$ is locally paracompact (Chapter IX, § 4, Exercise 27), and if there exists in $X$ a sequence of points $(x_n)$ which converges to a point distinct from all the $x_n$, then, $X$ must be locally compact [use Exercise 10 b) and Chapter IX, § 4, Exercise 25 c)].

c) Deduce from b) that if $X$ is metrizable but not locally compact, then there are points of $C_c(X; \mathbf{R})$ which have no countable fundamental system of neighbourhoods [use Exercise 9 a)]. Give a direct proof, using Exercise 5 a).