Conditions on Homeomorphism for a T1 Space

Theorem

Supposeis onto whereis aspace. A necessary and sufficient condition forto be a homeomorphism is

1.for every

or

2.

Proof

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.