Proof That the Homeomorphism Relation is Reflexive, Symmetric and Transitive
Theorem
If
and
are homeomorphic topological spaces then
and
Proof
Obviously
since for any topological space
the identity function
is a homeomorphism and X is homeomorphic to itself.
If
then a homeomorphism
exists from
to
Since
is a homeomorphism it is one to one and onto so
exists and is well defined and
is one to one and onto. Hence the morphism property is symmetric.
Ifthen a homeomorphismexists fromto
and if
then a homeomorphism
exists fromto
Composition of one to one and onto functions gives a one to one and onto function so
is 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.
