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 requires the existence of a countable set with certain properties, while without it such sets might not exist.
Important countability axioms for topological spaces:
- sequential space: a set is open if every sequence converging 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 subspace
- Lindelöf space: every open cover has a countable subcover
- σ-compact space: there exists a countable cover by compact 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:
