Proof That the Homeomorphism Relation is Reflexive, Symmetric and Transitive
Ifandare homeomorphic topological spaces thenand
Obviouslysince for any topological spacethe identity functionis a homeomorphism and X is homeomorphic to itself.
Ifthen a homeomorphismexists fromtoSinceis a homeomorphism it is one to one and onto soexists and is well defined andis one to one and onto. Hence the morphism property is symmetric.
Ifthen a homeomorphismexists fromtoand ifthen a homeomorphismexists fromtoComposition of one to one and onto functions gives a one to one and onto function sois one to one and onto, so is a homeomorphism. The homeomorphism property is transitive.
The homemorphism property is an equivalence relation since the reflexive, symmetric and transitive properties are satisfied.