Definition:Sequentially Compact Space

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

A subspace $$A \subseteq X$$ is sequentially compact in $$X$$ iff every infinite sequence in $$X$$ has a subsequence which converges to a point in $$X$$.

Sequentially Compact in Itself
A subspace $$A \subseteq X$$ is sequentially compact in itself iff every infinite sequence in $$A$$ has a subsequence which converges to a point in $$A$$.

This is understood to mean that $$A$$ is sequentially compact when we consider it as a topological space with the induced topology of $$X$$.

Clearly we can take $$A = X$$, in which case we can just refer to $$X$$ itself as being sequentially compact.

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 in the usual topology of $$\R^N$$.