Talk:Quasiuniformity Induces Topology

Counterexamples in Topology states that:
 * Every quasiuniformity $\UU$ on a set $X$ yields a topology $T$ on $X$ by taking as a neighborhood system for $X$ the sets $\map u x$ ...

So $\tau$ should not be defined as $\set {\map u x: u \in \UU: x \in X}$, but instead the topology generated from it.

(Indeed the set above is not a topology: (O2): $\exists u, x: \map u x = \map {\Delta_X} a \cap \map {\Delta_X} b = \O$ for $a \ne b$ contradicts (U1))

However, this also suggests that the topology induced by the quasiuniformity is the topology induced by the neighborhood system consisting of the $\map u x$, so what are we trying to prove then?

--RandomUndergrad (talk) 10:48, 8 June 2020 (EDT)