A **uniform space** (*X*, Φ) is a set *X* equipped with a nonempty family Φ of subsets of the Cartesian product *X* × *X* (Φ is called the **uniform structure** or **uniformity** of *X* and its elements **entourages** (French: neighborhoods or *surroundings*)) that satisfies the following axioms:

- if
*U*is in Φ, then*U*contains the diagonal Δ = { (*x*,*x*) :*x*∈*X*}. - 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*. - if
*U*is in Φ, then*U*-1 = { (*y*,*x*) : (*x*,*y*) in*U*} is also in Φ

If the last property is omitted we call the space **quasiuniform**.

One usually writes *U*={*y* : (*x*,*y*)∈*U*}. On a graph, a typical entourage is drawn as a blob surrounding the "*y*=*x*" diagonal; the *U*’s are then the vertical cross-sections. If (*x*,*y*) ∈ *U*, one says that *x* and *y* are *U-close*. Similarly, if all pairs of points in a subset *A* of *X* are *U*-close (i.e., if *A* × *A* is contained in *U*), *A* is called *U-small*. An entourage *U* is *symmetric* if (*x*,*y*) ∈ *U* precisely when (*y*,*x*) ∈ *U*. The first axiom states that each point is *U*-close to itself for each entourage *U*. The third axiom guarantees that being "both *U*-close and *V*-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage *U* there is an entourage *V* which is "half as large". Finally, the last axiom states the essentially symmetric property "closeness" with respect to a uniform structure.

A **fundamental system of entourages** of a uniformity Φ is any set **B** of entourages of Φ such that every entourage of Ф contains a set belonging to **B**. Thus, by property 2 above, a fundamental systems of entourages **B** is enough to specify the uniformity Φ unambiguously: Φ is the set of subsets of *X* × *X* that contain a set of **B**. Every uniform space has a fundamental system of entourages consisting of symmetric entourages.

The right intuition about uniformities is provided by the example of metric spaces: if (*X*,*d*) is a metric space, the sets

form a fundamental system of entourages for the standard uniform structure of *X*. Then *x* and *y* are *U*_{a}-close precisely when the distance between *x* and *y* is at most *a*.

A uniformity Φ is *finer* than another uniformity Ψ on the same set if Φ ⊇ Ψ; in that case Ψ is said to be *coarser* than Φ.

Read more about Uniform Space: Topology of Uniform Spaces, Uniform Continuity, Completeness, Examples, History

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—Peter Blos (20th century)

“The Federal Constitution has stood the test of more than a hundred years in supplying the powers that have been needed to make the Central Government as strong as it ought to be, and with this movement toward *uniform* legislation and agreements between the States I do not see why the Constitution may not serve our people always.”

—William Howard Taft (1857–1930)