Proof That a One to One Continuous Function From a Compact Space into a Hausdorff Space is a Homeomorphism
Theorem
Let
be one to one continuous function
from a compact space
into a Hausdorff space
is homeomorphic to
Proof
is a continuous bijection, hence one to one and onto. Hence
exists and is well defined.
is continuous if, for every closed subset
of
is a closed subset of
Letbe a closed subset of a compact space, thenis compact. Sinceis continuous
is a compact subset of
is Hausdorff, since it is a subspace of a Hausdorff space, henceis closed andis continuous.
Henceis a homeomorphism.
