Uniform space

In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. Uniform spaces generalize metric spaces and topological groups and therefore underlie most of analysis. They were introduced by Bourbaki.

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:

if U is in Φ, then U contains { (x, x) : x in X }.
if U is in Φ, then { (y, x) : (x, y) in U } is also in Φ
if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
if U and V are in Φ, then U ∩ V is in Φ
if U is in Φ, then there exists V in Φ such that, whenever (x, \y) and (y, z) are in V, then (x, z) is in U.

The set X together with a uniform structure Φ is called a uniform space. The elements of Φ are called entourages.

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:

every point is close to itself
if x is close to y, then y is close to x
relaxing a degree of closeness yields another degree of closeness
by combining two degrees of closeness, you get another one
to every degree of closeness, there exists another one that captures "twice as close".

The essential difference between a topological space and a uniform space is that in a uniform space, you can formalize the idea that "x₁ is about as far away from x₂ as y₁ is from y₂" while in a topological space you can only formalize "x₁ is about as far away from x as x₂ is from x".

Uniform spaces may be defined alternatively and equivalently using systems of pseudo-metrics, 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 T₀-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.