First axiom of countability

A concept in set-theoretic topology. A topological space satisfies the first axiom of countability if the defining system of neighbourhoods of every point has a countable base. The class of spaces satisfying the first axiom of countability was defined by F. Hausdorff (1914). All metric spaces, the space of continuous functions on a segment, etc., belong to this class. Spaces that satisfy the second axiom of countability also satisfy the first one. The converse is false, e.g. an uncountable space with the discrete topology does not satisfy the second axiom of countability.