, 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:
- 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.