A gauge of pseudometrics and, on a nonempty set, the uniformity it generates

A gauge of pseudometrics on $X$ is a family $\mathcal P$ of pseudometrics (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric). For finite $F\subseteq\mathcal P$ and $\varepsilon>0$, put $E(F,\varepsilon)=\{(x,y):p(x,y)<\varepsilon\text{ for every }p\in F\}$. If $X\ne\varnothing$, these sets form a filter base and generate a uniformity (Filter base and the filter it generates, The upward closure of a filter base is the smallest filter containing it), called the uniformity generated by $\mathcal P$.

For an already given uniformity $\mathcal U$ on $X$, a pseudometric $p$ is uniformly continuous for $\mathcal U$ when

$$\{(x,y):p(x,y)<\varepsilon\}\in\mathcal U$$

for every $\varepsilon>0$. With this terminology, each member of a gauge is uniformly continuous for the uniformity generated by that gauge.