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Important countability axioms for Topological Space s:
  • Sequential Space s: a set is open if every sequence converging to a point in the set is eventually in the set,

  • ,

  • ,

  • ,

  • has a countable subcover,

  • σ-compact Spaces : there exists a countable cover by compact spaces,


These axioms are not all unrelated. In particular, every second-countable space is first-countable, separable, and Lindelöf. Also, every σ-compact space is Lindelöf. For Metric Space s, first-countability is automatic, and second-countability, separability, and the Lindelöf property are all equivalent.

Other examples: