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 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.
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