Axiom of countability

Important examples

• 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
• 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

• 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