Definition:Sequentially Compact Space/In Itself

Definition
Let $T = \left({S, \tau}\right)$ be a topological space. A subspace $H \subseteq S$ is sequentially compact in itself iff every infinite sequence in $H$ has a subsequence which converges to a point in $H$.

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

Clearly we can take $H = S$, and refer to $S$ itself (or the space $T$) as being sequentially compact.