Uniform space

In the mathematical field of topology, a uniform space is a topological space 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. There are three equivalent definitions for a uniform space. They all consist of a space equipped with a uniform structure. This definition adapts the presentation of a topological space in terms of neighborhood systems. A nonempty collection of subsets of is a (or a ) if it satisfies the following axioms: If then where is the diagonal on If and then If and then If then there is some such that , where denotes the composite of with itself. The composite of two subsets and of is defined by If then where is the inverse of The non-emptiness of taken together with (2) and (3) states that is a filter on If the last property is omitted we call the space . An element of is called a or from the French word for surroundings. One usually writes where is the vertical cross section of and is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "" diagonal; all the different 's form the vertical cross-sections. If then one says that and are . Similarly, if all pairs of points in a subset of are -close (that is, if is contained in ), is called -small.