Uniform space

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In the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis.

In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "x is closer to a than y is to b" make sense in uniform spaces. By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A (i.e., in the closure of A), or perhaps that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone.