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.
This definition adapts the presentation of a topological space in terms of neighborhood systems.
is called a vicinity or entourage from the French word for surroundings.
On a graph, a typical entourage is drawn as a blob surrounding the "
Finally, the last axiom states that the property "closeness" with respect to a uniform structure is symmetric in
form a fundamental system of entourages for the standard uniform structure of
Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach that is particularly useful in functional analysis (with pseudometrics provided by seminorms).
can be shown to form a fundamental system of entourages of a uniformity.
A fundamental system of entourages of this uniformity is provided by the set of finite intersections of entourages of the uniformities defined by the individual pseudometrics
Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric.
A consequence is that any uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no.
that form a filter when ordered by star refinement.
Given a uniform space in the entourage sense, define a cover
Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods:
the following are equivalent: Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space.
A uniformity compatible with the topology of a completely regular space
can be defined as the coarsest uniformity that makes all continuous real-valued functions on
A fundamental system of entourages for this uniformity is provided by all finite intersections of sets
that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any
Indeed, as discussed above, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff.
In particular, if the topology of a vector space is Hausdorff and definable by a countable family of seminorms, it is metrizable.
In other words, a filter is Cauchy if it contains "arbitrarily small" sets.
Conversely, a uniform space is called complete if every Cauchy filter converges.
Complete uniform spaces enjoy the following important property: if
thus defined is in general not injective; in fact, the graph of the equivalence relation
Before André Weil gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using metric spaces.
Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in the book Topologie Générale and John Tukey gave the uniform cover definition.
Weil also characterized uniform spaces in terms of a family of pseudometrics.