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)

I haven't made the effort to think about either uniformities or quasiuniformities, Besides, Steen & Seebach are not good works to teach oneself from. As a result, I have done no work to analyse the statement from which this page was taken:

Every quasiuniformity $U$ on a set $X$ yields a topology $\tau$ on $X$ by taking as a neighborhood system for $X$ the sets $\map u x$ where $u \in U$ and $\map u x = \set {y: \tuple {x, y} \in u}$; there may be more than one quasiuniformity generating a given topology (Example $44$)...

so I crafted the page based on that, with a view to looking at it as and when I had the headspace to do so.

Feel free to take this on. --prime mover (talk) 15:42, 8 June 2020 (EDT)