Axiom of countability

Axiom of countability

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:

Relations:

  • Every first countable space is sequential.
  • Every second-countable space is first-countable, separable, and Lindelöf.
  • Every σ-compact space is Lindelöf.
  • A metric space is first-countable.
  • For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.

Other examples:

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