The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), the two characterizations of universal quotient maps below are mentioned.
Proposition. For a continuous map $p$, TFAE.
- $p$ is a universal quotient map (i.e., a regular epi that is stable under pullback).
- Fibers of adherence points of filters $\mathscr F$ contain adherence points of $p^\ast \mathscr F$ ($x$ is an adherence point of $\mathscr{F}$ if every neighborhood of $x$ has nonempty intersection with every element of $\mathscr{F}$).
- For every open cover $ \left\{ U_i \right\}_{i\in I}$ of a fiber $p^\ast \left\{ b \right\}$, there's a finite $I_0\subset I$ satisfying $b\in\operatorname{int} \left( p_\ast \bigcup_{i\in I_0}U_i\right)$.
I can't seem to find a geometrical way to think of either of these conditions, so I would like some help in finding intuition..
