Definition 1

A uniform structure (or uniformity) on a set $X$ is a structure given by a set $\mathfrak{U}$ of subsets of $X \times X$ which satisfies axioms $(\mathrm{F}_1)$ and $(\mathrm{F}_{\mathrm{II}})$ of Chapter I, § 6, no. 1 and also satisfies the following axioms:

(U₁) Every set belonging to $\mathfrak{U}$ contains the diagonal $\Delta$.

(U₂) If $V \in \mathfrak{U}$ then $\overline{V}^{-1} \in \mathfrak{U}$.

(U₃) For each $V \in \mathfrak{U}$ there exists $W \in \mathfrak{U}$ such that $W \circ W \subset V$.

!Figure 2

The sets of $\mathfrak{U}$ are called entourages of the uniformity defined on $X$ by $\mathfrak{U}$. A set endowed with a uniformity is called a uniform space.

If $V$ is an entourage of a uniformity on $X$, we may express the relation $(x, x') \in V$ by saying that "$x$ and $x'$ are $V$-close".