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*.

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".

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 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.