Definition:Sequentially Compact Space

Definition
Let $\left({X, \vartheta}\right)$ 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 $\R^N$.