Countability axioms is the common name used to refer to a set of properties of a topological space which have to do with the existence of countable sets, or countable families of open sets, satisfying certain conditions.
They are not axioms in the strict sense of the word, but they are usually named as such because one may think of them as additional basic properties that one can ask from a topological space.
- Second-Countable Space is First-Countable
- Second-Countable Space is Separable
- Second-Countable Space is Lindelöf
- Results about Countability Axioms can be found here.