Definition 2

A fundamental system of entourages of a uniformity is any set $\mathfrak{B}$ of entourages such that every entourage contains a set belonging to $\mathfrak{B}$.

Axiom (U_{III}) shows that if $n$ is any integer $> 0$ and $V$ runs through a fundamental system of entourages, then the sets $\overline{V}^n$ again form a fundamental system of entourages.

Entourages $V$ such that $V = \overline{V}^1$ are called symmetric. If $V$ is any entourage, then $V \cap \overline{V}^1$ and $V \cup \overline{V}^1$ are symmetric entourages, and axioms (F_{II}) and (U_{II}) show that the symmetric entourages form a fundamental system of entourages.

A set $\mathfrak{B}$ of subsets of $X \times X$ is a fundamental system of entourages of a uniformity on $X$ if and only if $\mathfrak{B}$ satisfies axiom (B_I) of Chapter I, § 6, no. 3, and also satisfies the following axioms:

(U'_I) Every set of $\mathfrak{B}$ contains the diagonal $\Delta$.

(U'_II) For each $V \in \mathfrak{B}$ there exists $V' \in \mathfrak{B}$ such that $V' \subset \overline{V}^1$.

(U'_III) For each $V \in \mathfrak{B}$ there exists $W \in \mathfrak{B}$ such that $\overline{W}^2 \subset V$.

If $X$ is not empty, a fundamental system of entourages of a uniform structure on $X$ is a base of the filter formed by the entourages of this structure (Chapter I, § 6, no. 3, Proposition 3).

(*) We recall (Set Theory, R, § 3, nos. 4 and 10) that if $V$ and $W$ are two subsets of $X \times X$, then the set of pairs $(x, y) \in X \times X$, such that $(x, z) \in W$ and $(z, y) \in V$ for some $z \in X$, is denoted by $V \circ W$ or $VW$, and that the set of pairs $(x, y) \in X \times X$ such that $(y, x) \in V$ is denoted by $\overline{V}^1$.

Examples of uniformities. \* 1) On the set $\mathbf{R}$ of real numbers we can define a uniformity, called the additive uniformity, as follows: for each $\alpha > 0$ let $V_\alpha$ be the subset of $\mathbf{R} \times \mathbf{R}$ consisting of all pairs $(x, y)$ such that $|x - y| < \alpha$; as $\alpha$ runs through the set of all real numbers $> 0$, the $V_\alpha$ form a fundamental system of entourages for the additive uniformity on $\mathbf{R}$. Similarly we can define a uniformity (again called the additive uniformity) on the set $\mathbf{Q}$ of rational numbers; we shall study these structures, and analogous uniform structures on groups, in Chapters III and IV. \* 2) Let $X$ be a set and let $R$ be an equivalence relation on $X$; let $C$ be the graph of $R$ in $X \times X$. Then $\Delta \subset C$ and $\overset{2}{C} = \overset{-1}{C} = C$; (Set Theory, R, § 5, no. 1); the set of subsets of $X \times X$ which consists of the set $C$ alone is therefore a fundamental system of entourages of a uniformity on $X$. In particular, if we take $R$ to be the relation of equality, then $C = \Delta$ and the entourages of the corresponding uniformity are all the subsets of $X \times X$ which contain $\Delta$; this uniformity is called the discrete uniformity on $X$, and the set $X$ endowed with this uniformity is called a discrete uniform space. 3) On the set $\mathbf{Z}$ of rational integers we can define a uniformity, important in the theory of numbers, as follows: given a prime number $p$, let $W_n$ be the set of all pairs $(x, y) \in \mathbf{Z} \times \mathbf{Z}$ such that $x \equiv y \pmod{p^n}$, for each integer $n > 0$. It is easily verified that the sets $W_n$ (for a fixed $p$) form a fundamental system of entourages of a uniformity on $\mathbf{Z}$, called the $p$-adic uniformity (cf. Chapter III, § 6, Exercises 23 ff., and Chapter IX, § 3, no. 2).

In accordance with the general definitions (Set Theory, R, § 8, no. 5), if $X$ and $X'$ are two sets each endowed with uniformities whose sets of entourages are $U$ and $U'$ respectively, then a bijection $f$ of $X$ onto $X'$ is an isomorphism of the uniformity of $X$ onto that of $X'$ if $g(U) = U'$, where $g = f \times f$.

For example, if $X$ and $X'$ are two equipotent sets, then every bijection of $X$ onto $X'$ is an isomorphism of the discrete uniformity of $X$ onto the discrete uniformity of $X'$.