Proof of Closed Sets Property for Homeomorphisms

Theorem

The following conditions are necessary and sufficient for a one to one mappingto be a homeomorphism.

1.for every

2.for every

Proof

is continuousfor everyandfor every(1)

Suppose now thatis a homeomorphism, thenis continuous and for every

Alsois continuous so from (1) we obtain

From the last two statements we obtain

Suppose that for everythen

By (1)is continuous. Also, for every

Henceis continuous andis a homeomorphism. Similarly for 2.