Definition:Sequentially Compact Space

Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.

Then $T$ is sequentially compact 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 $T$.

Clearly we can take $A = X$, in which case the main definition applies, and refer to $X$ itself (or the space $T$) as being sequentially compact.