Countability axioms in topology

Countability axioms in topology are properties that a topological space may satisfy which refer to the countability of certain structures within the space.

A separable space is one which has a countable dense subset.

A first countable space is one for which there is a countable filter base for the neighbourhood filter at any point.

A second countable space is one for which there is a countable base for the topology.

The second axiom of countability implies separability and the first axiom of countability. Neither implication is reversible.