# Category:First-Countable Spaces

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

A topological space $T = \struct {S, \tau}$ is **first-countable** or **satisfies the First Axiom of Countability** if and only if every point in $S$ has a countable local basis.

## Subcategories

This category has only the following subcategory.

## Pages in category "First-Countable Spaces"

The following 33 pages are in this category, out of 33 total.

### A

### C

- Compact Complement Topology is First-Countable
- Compact First-Countable Space is Sequentially Compact
- Condition for Open Extension Space to be First-Countable
- Countable Complement Space is not First-Countable
- Countable Product of First-Countable Spaces is First-Countable
- Countably Compact First-Countable Space is Sequentially Compact

### E

### F

- First-Countability is Hereditary
- First-Countability is not Continuous Invariant
- First-Countability is Preserved under Open Continuous Surjection
- First-Countable Space is Hausdorff iff All Convergent Sequences have Unique Limit
- First-Countable Space is Sequentially Compact iff Countably Compact
- Fortissimo Space is not First-Countable