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Axiom of countability

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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.

Contents

Important examples

Important countability axioms for topological spaces include:

  • 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 (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.

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

    References

    Axiom of countability Wikipedia
    (Text) CC BY-SA