I need some help with a topological proof, hopefully there is someone out there that can help.
Prove that a bijection f: X→Y is a homeomorphism if and only if f and f^-1 map closed sets to closed sets.
The definition of a homeomorphism is a map which is bijective continuous and open (maps open sets to open sets).
Assume is a homeomorphism.
is continuous and open so is continuous and open. But then is closed (maps closed sets to closed sets) and so is since let be a closed subset of . Then is open in and so is open in . But and therefore is closed in . Same argument shows is a closed map.
Now assume that and are bijective maps which take closed sets to closed sets. We need to show that is continuous and open. Consider an arbitrary open set . is closed and is closed in . Therefore is open in . So is continuous. Similar argument shows is open.