Sub-basis for Uniformity on Real Number Line

Theorem
Let $\struct {\R, \tau_d}$ be the real number line considered as a topological space under the usual (Euclidean) topology.

Let $a, b \in \R$ such that $a < b$.

Let $S_{a b}$ be the set of subsets of $\R$ defined as:
 * $S_{a b} = \set {\tuple {x, y}: x, y < b \text{ or } x, y > a}$

Then $S_{ab}$ is a basis for a uniformity $U$ which generates the usual topology on $\R$.

Note that $U$ is clearly not the usual metric uniformity.