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If $C(T,S)$ is the set of all continuos function between $T$ and $S$ metric spaces and $S$ compact with the uniform metric. What conditions are needed on $T$ and $S$ such that $C(T,S)$ be compact?

This is related with https://math.stackexchange.com/questions/1104044/compact-metric-space-implies-that-the-hyperspace-is-compact?noredirect=1#comment2251007_1104044 .

I would like some hint, not the complete answer.

Thanks!

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EQJ
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  • What is the topology on $C(T,S)$? The usual choice is the uniform topology, but $C(T,S)$ is typically not compact in this topology. – Nate Eldredge Jan 14 '15 at 16:32
  • In the asumptions $T$ and $S$ are compact. – EQJ Jan 14 '15 at 16:33
  • Doesn't help. For example, $C([0,1],[0,1])$ is not compact in the uniform topology; if $f_n(x) = x^n$ then ${f_n}$ is a sequence with no convergent subsequence. So I ask again: What is the topology on $C(T,S)$? – Nate Eldredge Jan 14 '15 at 16:35
  • I will edit the question. – EQJ Jan 14 '15 at 17:10
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    The edit doesn't help address Nate's question. To talk about $C(T,S)$ being compact, you need to put a topology on it. When $T$ and $S$ are metric spaces (or I guess uniform spaces), the uniform topology is a natural choice, which is why he mentioned it. What topology do you want to use? –  Jan 14 '15 at 17:16
  • I edit again. The topology is the uniform topology. – EQJ Jan 14 '15 at 17:19

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