AXIOM OF COUNTABILITY

In mathematics, an 'axiom of countability' is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
Important countability axioms for topological spaces:

★ sequential space: a set is open if every sequence converging 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 subspace

★ Lindelöf space: every open cover has a countable subcover

★ σ-compact space: there exists a countable cover by compact spaces
Relations:

★ Every first countable space is sequential.

★ Every second-countable space is first-countable, separable, and Lindelöf.

★ Every σ-compact space is Lindelöf.

★ A metric space is first-countable.

★ For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.
Other examples:

★ sigma-finite measure spaces

★ lattices of countable type