Definition:Countability Axioms

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. Euclidean space $$\R^N$$ satisfies all of the axioms below, and the same happens for many usual spaces, so they are properties that one may rely upon in many situations.

First-Countable Space
We say that a topological space $$(X,\vartheta)$$ is first-countable or satisfies the First Axiom of Countability if every point in $$X$$ has a countable neighborhood basis.

Second-Countable Space
We say that a topological space $$(X,\vartheta)$$ is second-countable or satisfies the Second Axiom of Countability if its topology has a countable base.

Separable Space
We say that a topological space $$(X,\vartheta)$$ is separable if there exists a countable dense subset of $$X$$.