A standard result in topology is that if $(K,\mathcal{T})$ is a compact space, and $f$ a continuous surjective map into another space $L$, then $L$ is compact.
I'm curious about what happens when we tweak the conditions a bit. Suppose $(S,d)$ and $(T,\delta)$ are metric spaces, with $(T,\delta)$ compact. Is there a known result of a necessary and sufficient condition for a subset $A\subset\mathcal{C}(S,T)$ to be compact also? Thanks for any proof or reference.
Here $\mathcal{C}(S,T)$ is the set of continuous functions from $S$ into $T$ with the standard metric $\rho(f,g)=\sup_{s\in S}\delta(f(s),g(s))$.