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 numerical function f continuous in $X \times Y$, the mapping $y \mapsto f(., y)$ from Y into $C(X; \mathbf{R})$ is continuous.

a) Let $y_0$ be a point, A a countable infinite subset of X, closed in X and all of whose points are isolated, I a bounded open interval in $\mathbf{R}$. Prove that, for the topology $\mathscr{T}$ the set $T(A, I)$ (X, p. 49, Exercise 10) has no interior point. (If $u_0 \in T(A, I)$, form a continuous mapping f from $X \times Y$ into $\mathbf{R}$ which is such that $f(., y_0) = u_0$ and that for every $z_n$, one has $f(., z_n) \notin T(A, I)$).

b) Deduce that if, in addition, for the topology $\mathscr{T}$, the mapping $(u, x) \mapsto u(x)$ from $C(X; \mathbf{R}) \times X$ into $\mathbf{R}$ is continuous, if X is locally paracompact (IX, p. 111, Exercise 36), and if there exists in X a sequence $(x_n)$ of points which converges towards a point distinct from the $x_n$, then X is necessarily locally compact (use Exercise 10 b), as well as IX, p. 110, Exercise 35 c)).

c) Conclude from b) that if X is metrizable and not locally compact, there exist points of $C_c(X; \mathbf{R})$ which do not admit a countable fundamental system of neighbourhoods (use X, p. 49, Exercise 9 a)). Give a direct proof of this fact using X, p. 48, Exercise 5 a).