Definition:Compatible Quasiuniformities

Definition
Let $\mathcal U_1$ and $\mathcal U_2$ be quasiuniformities on a set $S$.

Let $\left({\left({S, \mathcal U_1}\right), \tau_1}\right)$ and $\left({\left({S, \mathcal U_2}\right), \tau_2}\right)$ be the quasiuniform spaces generated by $\mathcal U_1$ and $\mathcal U_2$.

Then $\mathcal U_1$ and $\mathcal U_2$ are compatible (with each other) their topologies are equal.

That is, $\tau_1 = \tau_2$.