View source for 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|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.