title: compactnesses in metric-metrizable spaces • If a topological space is metrizable, then compactness, limit point compactness, and [sequential compactness](sequentially compact) are equivalent • A metric space is compact if and only if it is a complete metric space and is also totally bounded (JRM:Top.2 §45.1) • Proof sketch. Construct some infinite sequences of points. Since is totally bounded, then it is covered by a finite number of balls of size 1. One of these balls contains an infinite number of points of our sequence. Consider just the tail within that ball. Cover by balls of size 1/2; again, one of these balls contains an infinite number of points of our tail. Consider just that tail of the tail. Repeat infinitely and we have constructed a subsequence which is Cauchy. Since is complete, then this subsequence converges. So is sequentially compact, so it is compact. idk.