Proof That a One to One Continuous Function From a Compact Space into a Hausdorff Space is a Homeomorphism
Letbe one to one continuous functionfrom a compact spaceinto a Hausdorff space
is homeomorphic to
is a continuous bijection, hence one to one and onto. Henceexists and is well defined.
is continuous if, for every closed subsetofis a closed subset of
Letbe a closed subset of a compact space, thenis compact. Sinceis continuous is a compact subset ofis Hausdorff, since it is a subspace of a Hausdorff space, henceis closed andis continuous.
Henceis a homeomorphism.