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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!

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    Compactness of the *range* $T$ doesn't help *that* much for getting compactness in $C(S,T)$. To get a manageable condition for compactness in $C(S,T)$ you also need some sort of (local) compactness in the *domain* $S$. There are generalizations for non-locally compact $S$, which you can find in Kelley's General topology.2011-10-11
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    ^ Thanks, I checked out that reference and it seems Kelley generalizes the domain to $k$-spaces with no reference to the codomain being compact. Am I left to assume that the range being compact gives nothing and that the conditions for this particular problem are the same as the ones stated by Kelley?2011-10-11
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    I am having this wicked deja-vu, didn't you ask this question already a few days ago or so?2011-10-11
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    @Asaf: [you're still sane](http://math.stackexchange.com/questions/71283/how-to-generalize-arzela-ascoli) -- well, sort of :) .@Dustin: instead of deleting and re-posting your question, you can edit the old question. Then it will be "bumped" to the front page.2011-10-11
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    oh! that makes sense. I wasn't sure if "bumping" worked on a stackexchange site. I actually linked to that in my post as well. But anyways, I'm still stuck on the problem.. I've been trying different conditions and seeing if they fit, but none seem to work.2011-10-11
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    @t.b.: I would not go as far as declaring me "sane" all of a sudden, but it is good to know that the old brain in a box still works. :-)2011-10-11

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