Theorem
The following conditions are necessary and sufficient for a one to one mapping
to be a homeomorphism.
1.
for every
2.
for every
Proof
is continuous
for every
and
for every(1)
Suppose now thatis a homeomorphism, thenis continuous and for every
Also
is 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.