Definition:Sequentially Compact Space

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.


Then $H$ is sequentially compact in $T$ if and only if every infinite sequence in $H$ has a subsequence which converges to a point in $S$.


Sequentially Compact in Itself

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 $T$.


Also see

  • Results about sequentially compact spaces can be found here.

Internationalization

Sequentially Compact is translated:

In Dutch: rijcompact


Sources