Definition:Sequentially Compact Space
Jump to navigation
Jump to search
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 $7.2.1$