# Compact = Closed + Bounded + (?)

In $\mathbb{R}^n$ we know (Heine-Borel Theorem) that a set is compact if and only if it is closed and bounded.

In $C(X)$ for a compact metric space $X$, we know (corollary of Ascoli-Arzela Theorem) that a set is compact if and only if it is closed, bounded, and equicontinuous.

I am looking for as many examples as I can of other spaces where the extra condition for compactness is known.

Also, I am looking for as many examples as I can of (important) spaces where the extra condition is not currently known.

I am planning on doing some research (under a professor) and I thought this topic was particularly interesting, so I would very much appreciate some examples to start off with, just so I can get a feel for the problem.