nLab topological submersion

A topological submersion is a map in Top generalising the sort of map that is called a submersion in Diff.

There are two definitions of a topological submersion $p\colon Y \to X$:

• Each point in $Y$ has a neighbourhood $U$ such that $p\big|_U\colon U \simeq p(U) \times Z \to p(U)$ is projection on the first factor. Sometimes $Z$ is required to be a cartesian space $\mathbb{R}^n$, but this is a bit restrictive.

• Each point $p$ of $Y$ has a local section $\sigma\colon V \to Y$ with $x\in V$ and $p = \sigma(x)$.

The second definition includes the first as a special case.

Surjective topological submersions form a singleton Grothendieck pretopology on Top.