\begin{definition}[Definition:Sequentially Compact Space/In Itself]
Let $T = \struct {S, \tau}$ 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.
\end{definition}