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:YXp\colon Y \to X:

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

  • Each point pp of YY has a local section σ:VY\sigma\colon V \to Y with xVx\in V and p=σ(x)p = \sigma(x).

The second definition includes the first as a special case.

Surjective topological submersions form a singleton Grothendieck pretopology on Top.

Last revised on August 24, 2011 at 10:17:46. See the history of this page for a list of all contributions to it.