Let $X, Y$ be topological spaces and $C(X,Y)$ the set of continuous functions with the compact open topology defined as having the subbasis consisting of all subsets $C(K, U)$ of functions $f$ for which $f(K) \subseteq U$, where $K$ is compact and $U$ is open. My question has two aspects:
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I am particularly interested in the case where $X,Y$ are both second countable, locally euclidian and Hausdorff. Is then $C(X,Y)$ second countable, or at least first countable?
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Are there any known weaker sufficient conditions on $X$ and $Y$ such that $C(X,Y)$ is second countable or at least first countable?
Regarding 1., I'm thinking $X$ being locally euclidian is useful because given a basis, the sets in it which are relatively compact form a subbasis. But before I can get to that, I am already stuck at the fact that, if $U$ is open in $Y$, $\left\{ U_i \right\}$ is a countable basis and $U=\cup_{i \in I_U} U_i$, one only has $\cup_{i \in I_U} C(K, U_i) \subseteq C(K, U)$, but not necessarily equality.
Best Answer
Locally Euclidean as such is irrelevant. But it is well-known that if $X$ and $Y$ are second countable Hausdorff and $X$ is locally compact (which does follow from locally Euclidean), then $C(X,Y)$ is second countable.
If $C(X,Y)$ is second countable, $Y$ also is (as the set of constant functions is a homeomorphic copy of $Y$ in it, and second countability is hereditary). For a locally compact $X$ and $Y=\Bbb R$ Thm 4.8 in this survey paper shows that $C(X,Y)$ second countable implies $X$ second countable too. I think this will stay true for arbitrary second countable metrisable $Y$.
As to first countability, this is more complicated IIRC. McCoy and Ntantu have written papers on the problem of when $C(X,Y)$ is first countable. The linked paper I gave has some references to get you started on that if you're interested.