# Category:Second-Countable Spaces

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This category contains results about **Second-Countable Spaces**.

A topological space $T = \struct {S, \tau}$ is **second-countable** or **satisfies the Second Axiom of Countability** if and only if its topology has a countable basis.

## Subcategories

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

## Pages in category "Second-Countable Spaces"

The following 44 pages are in this category, out of 44 total.

### C

- Cantor Space is Second-Countable
- Compact Complement Topology is Second-Countable
- Condition for Open Extension Space to be Second-Countable
- Countable Closed Ordinal Space is Second-Countable
- Countable Discrete Space is Second-Countable
- Countable Excluded Point Space is Second-Countable
- Countable Fort Space is Second-Countable
- Countable Open Ordinal Space is Second-Countable
- Countable Product of Second-Countable Spaces is Second-Countable

### E

### M

### R

### S

- Second-Countability is Hereditary
- Second-Countability is not Continuous Invariant
- Second-Countability is Preserved under Open Continuous Surjection
- Second-Countable Space is Compact iff Countably Compact
- Second-Countable Space is First-Countable
- Second-Countable Space is Lindelöf
- Second-Countable Space is Separable
- Second-Countable T3 Space is T5
- Separable Metric Space is Second-Countable
- Sequentially Compact Metric Space is Second-Countable

### U

- Uncountable Closed Ordinal Space is not Second-Countable
- Uncountable Discrete Space is not Second-Countable
- Uncountable Excluded Point Space is not Second-Countable
- Uncountable Open Ordinal Space is not Second-Countable
- Uncountable Particular Point Space is not Second-Countable
- Uncountable Product of Second-Countable Spaces is not always Second-Countable
- Urysohn's Metrization Theorem