Definition:Sequentially Compact Space/In Itself

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Let $T = \struct {S, \tau}$ be a topological space.

A subspace $H \subseteq S$ is sequentially compact in itself if and only if 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.

Also see

  • Results about sequentially compact spaces can be found here.