compact-open metrizability

See:

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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