Let $C(X,Y)$ be the space of continuous functions on two metric spaces $X$ and $Y$. If $\mathcal{F}$ is a family of functions of $C(X,Y)$ are there any conditions on $X$ or $Y$ such that an analogous formation of Arzela Ascoli shows that $\mathcal{F}$ is precompact(compact if closed).
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Sure. You can prove a version of the theorem that says that $\mathcal{F}$ is compact if and only if $\mathcal{F}$ is equicontinuous and $$ \overline{\{f(x) : f \in \mathcal{F}\}} \subseteq Y \text{ is compact for each } x \in X. $$ The space $X$ can even be swapped for a locally compact Hausdorff space. You can take a look here for references.
