Axiom of countability

Important countability axioms for topological spaces include: • sequential space: a set is closed if and only if every convergent sequence in the set has its limit point 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==