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.
Important examples
Important countability axioms for topological spaces include:[1]
- 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.
Related concepts
Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type.
References
- ^ Nagata, J.-I. (1985), Modern General Topology, North-Holland Mathematical Library (3rd ed.), Elsevier, p. 104, ISBN 9780080933795.