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If X is a set, a nonempty system Φ of subsets of the Cartesian product X × X is called a uniform structure on X if the following axioms are satisfied:
Intuitively, two points x and y are "close together" if the pair (x, y) is contained in many entourages. A single entourage captures a particular degree of "closeness". Interpreted thusly, the axioms mean the following:
Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach which is often useful in functional analysis.
Every uniform space X becomes a topological space by defining a subset O of X to be open if and only if for every x in O there exists an entourage V such that { y in X : (x, y) in V } is a subset of O. It is possible that two different uniform structures generate the same topology on X.
Every metric space (M, d) can be considered as a uniform space by defining a subset V of M × M to be an entourage if and only if there exists an ε > 0 such that for all x, y in M with d(x, y) < ε we have (x, y) in V. This uniform structure on M generates the usual topology on M.
Every topological group (G,*) becomes a uniform space if we define a subset V of G × G to be an entourage if and only if the set {x*y^{1} : (x, y) is in V} is a neighborhood of the identity element of G. This uniform structure on G is called the right uniformity on G, because for every a in G, the right multiplication x > x*a is uniformly continuous with respect to this uniform structure. One may also define a left uniformity on G; the two need not coincide, but they both generate the given topology on G.
Every uniform space is a completely regular topological space, and conversely, every completely regular space can be turned into a uniform space (often in many ways) so that the induced topology coincides with the given one.
A uniform space X is a T_{0}space if and only if the intersection of all the elements of its uniform structure equals the diagonal {(x, x) : x in X}. If this is the case, X is in fact a Tychonoff space and in particular Hausdorff.
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