Definition 3

If $(f_i)_{i \in I}$ is a family of pseudometrics on a set $X$, then the least upper bound of the set of uniformities defined on $X$ by the pseudometrics $f_i$ is called the uniformity defined by the family $(f_i)$.

Two families of pseudometrics on $X$ are said to be equivalent if they define the same uniformity on $X$.

From the definition of the least upper bound of a set of uniformities (Chapter II, § 2, no. 5), the filter of entourages of the uniformity $U$ defined on $X$ by a family of pseudometrics $(f_i)_{i \in I}$ is the filter generated (Chapter I, § 6, no. 2) by the family of sets $f_i^{-1}([0, a])$, where $i$ runs through $I$ and $a$ runs through the set of real numbers $> 0$. In other words, we obtain a fundamental system of entourages of $\mathscr{U}$ by proceeding as follows: we take at random a finite number of indices $i_1, i_2, \ldots, i_n$ and, corresponding to each $i_k$, a number $a_k > 0$; then we consider the set $V$ of pairs $(x, y) \in X \times X$ such that $f_{i_k}(x, y) \leq a_k$ for $1 \leq k \leq n$; these sets $V$ (for all possible choices of $n$, the $i_k$ and the $a_k$) form a fundamental system of entourages for $\mathscr{U}$. Moreover, we may restrict ourselves to the case in which all the $a_k$ are equal to the same number $a > 0$, since the entourage consisting of all pairs $(x, y)$ such that

is evidently contained in $V$.

For each finite subset $H$ of $I$, let $g_H$ denote the upper envelope of the family $(f_i)_{i \in H}$. As $H$ runs through the set of all finite subsets of $I$ and $a$ runs through the set of real numbers $> 0$, the sets $g_H^{-1}([0, a])$ form a fundamental system of entourages of the uniformity $\mathscr{U}$. Now the $g_H$ are pseudometrics on $X$ (no. 1), and the upper envelope of a finite number of functions of the family $(g_H)$ belongs to this family, by definition; we express this property by saying that the family of pseudometrics $(g_H)$ is saturated. The family of pseudometrics $(g_H)$ is therefore equivalent to the family $(f_i)$, and is said to be the family of pseudometrics obtained by saturating $(f_i)$. From what has just been said it follows that we may always restrict ourselves to considering uniformities defined by saturated families of pseudometrics.

In the particular case where $I$ is a finite set, this argument shows that the uniformity defined by the family of pseudometrics $(f_i)_{i \in I}$ is also defined by the single pseudometric $g = \sup_{i \in I} f_i$.

Let $\mathscr{U}, \mathscr{U}'$ be two uniformities on $X$, defined respectively by two saturated families $(f_i)_{i \in I}, (g_x)_{x \in K}$. Then $\mathscr{U}$ is coarser than $\mathscr{U}'$ if and only if, for each index $i \in I$ and each real number $a > 0$, there is an index $x \in K$ and a number $b > 0$ such that the relation $g_x(x, y) \leq b$ implies $f_i(x, y) \leq a$.

Example of a uniformity defined by a family of pseudometrics. Let $(f_i)_{i \in I}$ be an arbitrary family of (finite) real-valued functions defined on a set $X$. Let $\mathscr{U}$ be the coarsest uniformity on $X$ with respect to which the $f_i$ are uniformly continuous (Chapter II, § 2, no. 3). Then it follows from the definition of the entourages of $\mathscr{U}$ (loc. cit.) that $\mathscr{U}$ is the uniformity defined on $X$ by the pseudometrics