Axiom of countability

In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) 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:[1]

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.

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


  1. Nagata, J.-I. (1985), Modern General Topology, North-Holland Mathematical Library (3rd ed.), Elsevier, p. 104, ISBN 9780080933795.
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