title: compactnesses in metric-metrizable spaces • If a topological space $X$ is metrizable, then compactness, limit point compactness, and [sequential compactness](sequentially compact) are equivalent • A metric space $(X, d)$ is compact if and only if it is a complete metric space and is also totally bounded (JRM:Top.2 §45.1) • Proof sketch. $(\Leftarrow)$ Construct some infinite sequences of points. Since $X$ 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 $X$ 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 $X$ is complete, then this subsequence converges. So $X$ is sequentially compact, so it is compact. $(\Rightarrow)$ idk.