Definition:Compatible Quasiuniformities

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

Let $\left({\left({X, \mathcal U_1}\right), \vartheta_1}\right)$ and $\left({\left({X, \mathcal U_2}\right), \vartheta_2}\right)$ 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) iff their topologies are equal, i.e. iff $\vartheta_1 = \vartheta_2$.