Definition:Sequentially Compact Space/In Itself
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Definition
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.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Definitions $7.2.1$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties