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Condition for family of continuous maps to be compact?
I was reading through general-topology posts and I couldn't quite get this one. Here's a reformulation of the question:
Suppose $(S,d)$ and $(T,\delta)$ are metric spaces where $(T,\delta)$ is compact. Denote $C(S,T)$ to be the set of continuous functions $S\rightarrow T$ with the metric $\rho(f,g)=\sup_{s\in S}\delta(f(s),g(s))$.
Is there a (necessary and sufficient) condition for a subset $A\subset C(S,T)$ to be compact?
This seems like a generalization of Arzela-Ascoli, but I'm not exactly sure how it plays out. Any help would be greatly appreciated!