# Category:Sequentially Compact Spaces

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This category contains results about **Sequentially Compact Spaces**.

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 $S$.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Sequentially Compact Spaces"

The following 32 pages are in this category, out of 32 total.

### C

- Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Sequentially Compact
- Compact Complement Topology is Sequentially Compact
- Compact Subspace of Metric Space is Sequentially Compact in Itself
- Compactness and Sequential Compactness are Equivalent in Metric Spaces
- Complete and Totally Bounded Metric Space is Sequentially Compact
- Composition of Continuous Mapping on Compact Space Preserves Uniform Convergence
- Countable Product of Sequentially Compact Spaces is Sequentially Compact
- Countably Compact First-Countable Space is Sequentially Compact
- Countably Compact Metric Space is Sequentially Compact

### E

### F

### M

### S

- Sequential Compactness is Preserved under Continuous Surjection
- Sequentially Compact Metric Space is Compact
- Sequentially Compact Metric Space is Complete
- Sequentially Compact Metric Space is Lindelöf
- Sequentially Compact Metric Space is Second-Countable
- Sequentially Compact Metric Space is Separable
- Sequentially Compact Metric Space is Totally Bounded
- Sequentially Compact Metric Subspace is Sequentially Compact in Itself iff Closed
- Sequentially Compact Space is Countably Compact
- Subset of Indiscrete Space is Compact and Sequentially Compact
- Subset of Indiscrete Space is Sequentially Compact