Let $X$ and $Y$ be topological groups.
The space $\mathrm{Cont}(X,Y)$ of continuous functions will be given the compact-open topology.
The subspace $\mathrm{ContHom}(X,Y)$ of continuous homomorphisms will be given the subspace topology.
Without assuming that $Y$ is a metric space, are there conditions on $X$ and $Y$ that imply that $\mathrm{ContHom}(X,Y)$ is locally compact?
One special case of particular importance is the fact that $\mathrm{ContHom}(G,T)$ is locally compact, where $G$ is a locally compact Hausdorff abelian group, and $T$ is the circle group.
For this special case, the standard proof uses equicontinuity and Arzela-Ascoli. But this approach only works if $Y$ is a metric space.