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Let $X$ be a topological space and $Y$ a topological group. Then $C(X,Y)$ is a group, and can also be endowed with the compact-open topology.

Is $C(X,Y)$ in the compact-open topology necessarily a topological group? If not, is there some property of $X$ which will guarantee it?

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    $\begingroup$ When $X$ is compactly-generated and Hausdorff, then $C(X,-)$ is right adjoint to $-\times X$, so preserves limits (possibly you need to restrict to $Y$ Hausdorff as well). Any functor that preserves limits sends group objects to group objects. If you are happy with not quite the compact-open topology, then you can just ask that $X$ is core-compact. See discussion at ncatlab.org/nlab/show/exponential+law+for+spaces $\endgroup$
    – David Roberts ♦
    Commented May 25, 2020 at 11:48
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    $\begingroup$ check out section 2.6 in Narici, Beckenstein - Topological vector spaces $\endgroup$
    – erz
    Commented May 25, 2020 at 14:22

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