Axiom of countability
# Axiom of countability

In mathematics, an axiom of countability is a property of certain mathematical objects 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:

* 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.

