The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman, the two characterizations of universal quotient maps below are mentioned.

Proposition. For a continuous map $p$, TFAE.

• $p$ is a universal quotient map.
• Fibers of adherence points of filters $\mathscr F$ contain adherence points of $p^\ast \mathscr F$.
• For every open cover $ \left\{ U_i \right\}_{i\in I}$ of a fiber $p^\ast \left\{ b \right\}$, theres 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..

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..