In the paper
R. Arens: A Topology for Spaces of Transformations, Ann. of Math. 47(1946), 480-495
the author states in the introduction that if $B$ is a metric space and the space of continuous functions $C(A,B)$ in the compact-open topology is metrizable, then $A$ is hemicompact.
Later on, Theorem 8 says: If $C(A,B)$ satisfies the first axiom of countability and if for all points $x,y \in A$ there is a continuous $f: A \to \mathbb{R}$ with $f(x) \neq f(y)$, then $A$ is hemicompact.
Now, if $C(A,B)$ is metrizable, it's first countable. But I don't know how to find the $f$.
Can anyone give a hint ?
Added: In light of David's comment, it's clear that Theorem 8 alone doesn't imply hemicompactness. Thus the question:
Is it true that if $B$ is a metric space and $C(A,B)$ is metrizable, then $A$ is hemicompact ?
From the quoted Theorem 8 it's clear that if $A$ has in addition the separation property by continuous functions, then it's hemicompact. But since the converse ($A$ hemicompact, $B$ metric $\Rightarrow$ $C(A;B)$ metrizable) is true without this separation property, I wonder if it's really necessary.
