Editing Uniform space (section)

{{Short description|Topological space with a notion of uniform properties}}{{No footnotes|date=May 2009}}

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