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