Proof That the Homeomorphism Relation is Reflexive, Symmetric and Transitive

Theorem

Ifandare homeomorphic topological spaces thenand

Proof

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.