Here I asked a question about $3$ definitions of quotients for uniform spaces being equivalent (uniform space here, is a Hausdorff space together with uniform covers - those can be found on wikipedia).
However, I'm stuck at the proof of point $7$ from Isbell's book "Uniform Spaces".
I don't understand the first part of this proof. That is, why is $f_0$ uniformly continuous.
If anyone could explain me it I'd be grateful!
The existence of $f_0$ as a function is clear from the condition that $f$ and $q$ satisfy together (any $z \in Q$ is of the form $z=q(x)$ for some $x \in X$ and we then define $f_0(z) = f(x)$ for that $x$, which does not really depend on the $x$ we pick by the aforementioned condition, $f_0 \circ q=f$ by construction).
If $Q$ gets the weak preuniformity $\mu_{f_0}$ induced by $f_0$, this means that for any uniform cover $\mathcal{U}$ of $Y$, $f_0^{-1}[\mathcal{U}]$ is in $\mu_{f_0}$, (and these form a generating set) so trivially $f_0$ is uniformly continuous. And their inverse images under $q$ are just $f^{-1}[\mathcal{U}]$ by the commutative property, so uniform in $X$.