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.