Apart from
>>11405852, given any functions , you may consider the map (here by I don't denote "the" inverse function, but a map into subsets of the domain of ) mapping any to the set .
E.g., with reference to your picture:
Those are "fibers" over Y and you can equip them with odd topologies to make for interesting "function spaces" (a function here is a choice of a point in each fiber.).
E.g. the set is a fiber over (hovering "over") .
Btw., the wish that every surjective function has a right inverse such that
(going up in the fiber and down, in a way that you end up where you started), is equivalent to the Axiom Of Choice: Saying there's a q means there's a selection of some point for each fiber.
