Axiom of countability

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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 spaces: a set is open if every sequence converging to a point in the set is eventually in the set,
• first-countable spaces: every point has a countable local base,
• second-countable spaces: the topology has a countable base,
• separable spaces: there exists a countable dense subspace,
• Lindelöf spaces: every open cover has a countable subcover,
• σ-compact spaces: there exists a countable cover by compact spaces,

These axioms are not all unrelated. In particular, every second-countable space is first-countable, separable, and Lindelöf. Also, every σ-compact space is Lindelöf. For metric spaces, first-countability is automatic, and second-countability, separability, and the Lindelöf property are all equivalent.

Other examples:

• sigma-finite measure spaces
• lattices of countable type