Exercise 2, § 1

Let $f$ be a mapping of $X \setminus X$ into $[0, +\infty]$. Then the family of sets $\{f(x, x') > s\}$, where $s$ runs through the set of all real numbers $> 0$, forms a fundamental system of entourages of a uniformity on $X$ if and only if $f$ satisfies the following conditions: i) $f(x, x) = 0$ for all $x \in X$; ii) for each $s > 0$ there exists $t > 0$ such that the relation $f(x, x') \leq t$ implies $f(x, x') \leq s;$ iii) for each $s > 0$ there exists $c > 0$ such that the relations $f(x, z) \leq c$ and $f(z, y) \leq c$ imply $f(x, y) \leq a$.

The conditions ii) and iii) are satisfied in particular if there exists a mapping $\varphi$ of $I = [0, +\infty]$ onto itself which is continuous and zero at the point 0, and a mapping $\psi$ of $I \setminus I$ into $I$ which is continuous and zero at the point $(0, 0)$, such that we have identically