Remark

A uniformity $\mathscr{U}$ on $X$ is defined by the family $\Phi$ of all pseudometrics on $X$ which are uniformly continuous on $X \times X$. For clearly the uniformity defined by the family $\Phi$ is coarser than $\mathscr{U}$; conversely, Theorem 1 shows that there is a subfamily of $\Phi$ which defines the uniformity $\mathscr{U}$ and therefore the uniformity defined by $\Phi$ is finer than $\mathscr{U}$.