Countability axioms in topology
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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.