Is $\operatorname{Hol}(S,T)$ locally compact?

Question

$\def\Hol{\operatorname{Hol}}$ $S$ and $T$ are compact Riemann surfaces, $\Hol(S,T)$ denotes all the holomorphic fuctions from $S$ to $T$, if we give the "locally uniform convergence" topology to $\Hol(S,T)$. How to prove $\Hol(S,T)$ is locally compact. I know this topology has a basis $\{N_{K,\varepsilon}(f)\}$,where $N_{K,\varepsilon}(f)=\{g|d(gx,fx)<\varepsilon,\forall x\in K\}$