Proof of Closed Sets Property for Homeomorphisms
The following conditions are necessary and sufficient for a one to one mappingto be a homeomorphism.
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.