# Axiom of countability

In mathematics, an **axiom of countability** is a property of certain mathematical objects (usually in a category) that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist.

## Important examples

Important countability axioms for topological spaces include:[1]

- sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set
- first-countable space: every point has a countable neighbourhood basis (local base)
- second-countable space: the topology has a countable base
- separable space: there exists a countable dense subset
- Lindelöf space: every open cover has a countable subcover
- σ-compact space: there exists a countable cover by compact spaces

## Relationships with each other

These axioms are related to each other in the following ways:

- Every first-countable space is sequential.
- Every second-countable space is first countable, separable, and Lindelöf.
- Every σ-compact space is Lindelöf.
- Every metric space is first countable.
- For metric spaces, second-countability, separability, and the Lindelöf property are all equivalent.

## Related concepts

Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type.

## References

- Nagata, J.-I. (1985),
*Modern General Topology*, North-Holland Mathematical Library (3rd ed.), Elsevier, p. 104, ISBN 9780080933795.

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