Definition:Sequentially Compact Space

Definition

Let $T = \struct {S, \tau}$ 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

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Definitions $7.2.1$