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Given topological spaces $X$ and $Y$ the set $C(X,Y)$ of all continuous functions $f:X\to Y$ becomes a topological space with the compact-open topology (that is the topology generated by the sets $C_{K,U}(X,Y)$ of those continuous functions satisfying $f(K)\subseteq U$, where $K$ ranges over the compact subsets of $X$ and $U$ ranges over the open subsets of $Y$).

It is a rather straightforward exercise that if $Y$ is metrizable and $X$ is hemicompact (meaning that there is a countable family of chosen compact sets in it such that every compact set is contained in one of the chosen ones) then the compact-open topology on $C(X,Y)$ is metrizable.

I'm curious to know how reversible this situation is. In more detail:

1) Is there a condition $C$ such that the following is true: If $Y$ is metrizable and the compact-open topology on $C(X,Y)$ is metrizable then $X$ has property $C$?

2) What would be a 'boundary' type example for a non-hemicompact space $X$ and a metrizable space $Y$ such that $C(X,Y)$ is metrizable?

3) What would be a 'boundary' type example for a non-hemicompact space $X$ and a metrizable space $Y$ such that $C(X,Y)$ is not metrizable?

By 'boundary' type examples I mean ones that illustrate the border line of the necessity of the hemicompact condition. So for 2) it would in some sense be a large space, showing how far one can get from being hemicompact yet still have the metrizability of the compact-open topology, while for 3) it would be in some sense a small space.


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Robert A. McCoy, Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Springer Lecture Notes in Math, Volume 1315 (1988), page 68,

for a metrizable space $Y$, the space of continuous functions $C(X,Y)$ (equipped with the compact-open topology) is metrizable iff $X$ is hemicompact.

In the book, they assume that $X, Y$ are completely regular Hausdorff and that $Y$ contains a nontrivial path, but maybe the latter requirement can be omitted in this theorem. (It is left as an exercise, but, I think, it follows from the prior results in the book.)

Hope it helps.

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