sequentially compact

Sequential compactness captures compactness through the eyes of sequences instead of covers: no matter how you wander through the space, taking step after step, you can always pause along an infinite subset of your steps that homes in on a single point. Nothing can escape to infinity or to a hole, because some subsequence is always forced to converge.

Formally, a space X is sequentially compact if every sequence in X has a subsequence converging to a point of X. This is the form of compactness an analyst meets first, since it is exactly the conclusion of the Bolzano–Weierstrass theorem applied to closed bounded subsets of R^n.

In general topological spaces, sequential compactness and (cover) compactness are logically independent — neither implies the other without extra assumptions. The reconciliation is metric spaces: in a metric space the two coincide, and there compact, sequentially compact, and limit point compact all mean the same thing. So the loose calculus slogan “compact means every sequence has a convergent subsequence” is true exactly where you usually use it, but is not the definition of compactness in full generality.