about Heine-Borel Theorem in a function space

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.