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In Pugh's real mathematical analysis. About the Heine-Borel Theorem in a function space, it states that a subset $\epsilon$ $\in C^0$ is compact if and only if it is closed, bounded, and equicontinuous. For $C^0$ in the book it is the continuous functions mapping from $[a,b]$ to $R$. I am not sure whether it holds for continuous functions mapping compact metric space $M$ to $M$, namely, $C^0[M,M]$?
It will be so useful if we can use it in that sense.

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  • $\begingroup$ I think it is also true and the proof are more or less the same. Not sure if one can find a proof though. $\endgroup$ – user99914 Feb 1 '15 at 4:49
  • $\begingroup$ @John I tried to find Heine Borel Theorem in a function space in wiki but I couldn't find it. $\endgroup$ – jack Feb 1 '15 at 4:51
  • $\begingroup$ That's more commonly called Arzela Ascoli theorem. But it is still not what you are looking for (but close). $\endgroup$ – user99914 Feb 1 '15 at 4:52
  • $\begingroup$ Indeed, it is probably best to keep the name «Heine-Borel» for the property that a set is compact iff it is closed and bounded. $\endgroup$ – Mariano Suárez-Álvarez Feb 1 '15 at 4:59
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    $\begingroup$ Rather than «use a theorem» you'll need to just follow the steps in the proof of the Ascoli-Arzela theorem, adapted to your situation. $\endgroup$ – Mariano Suárez-Álvarez Feb 1 '15 at 5:02

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