# 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