# 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.

For Banach spaces, there is a big table in Dunford & Schwartz characterizing compactness of subsets in many different spaces.

Zev is right that "complete + totally bounded" is the general formulation. For subset in Banach space, complete holds if and only if closed. So the question is to characterize total boundedness.

• Thanks, GEdgar! What is the book by Dunford & Schwarz?
– Tim
Nov 25, 2011 at 22:28
• Tim: Just Google "Dunford and Schwarz" (or, better, "Dunford and Schwartz") and you'll figure this out yourself.
– KCd
Nov 25, 2011 at 22:34
• @KCD: Thanks! Found it.
– Tim
Nov 25, 2011 at 22:39