Sub-basis for Uniformity on Real Number Line

Theorem
Let $\left({\R, \tau_d}\right)$ 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 $S_{ab}$ be the set of subsets of $\R$ defined as:
 * $S_{ab} = \left\{{\left({x, y}\right): x, y < b \text{ or } x, y > a}\right\}$

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.