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 :
Each point in has a neighbourhood such that is projection on the first factor. Sometimes is required to be a cartesian space , but this is a bit restrictive.
Each point of has a local section with and .
The second definition includes the first as a special case.
Surjective topological submersions form a singleton Grothendieck pretopology on Top.