Definition:Sequentially Compact Space

Definition
Let $$(X,\vartheta)$$ be a topological space.

We say that $$(X,\vartheta)$$ is sequentially compact if every sequence in $$X$$ has a convergent subsequence.

Relationship to compactness
Sequential compactness is a useful consequence of compactness, and is a frequently used property. In general, a compact space is sequentially compact, but the converse is not true. However, whenever the topology of the space is induced by a metric, sequential compactness is equivalent to compactness, and so sequential compactness is a characterization of compactness in many usual cases, notably for compactness in $$\R^N$$.