Conditions on Homeomorphism for a T1 Space
Supposeis onto whereis aspace. A necessary and sufficient condition forto be a homeomorphism is
A topological spaceis aspace if each singleton set is closed so thatfor eachWith this definition each metric space is aspace.
Sinceis a homeomorphism it is one to one andor
Now we show thatis one to one. Supposethen
Sinceis aspace,andis one to one. Hencefor everyandis a homeomorphism.