
Homeomorphism
Dear Colleagues,
Let $\displaystyle X$ and $\displaystyle Y$ be metric spaces, $\displaystyle X$ compact, and $\displaystyle T:X\longrightarrow Y$ bijective and continuous. Show that $\displaystyle T$ is a homeomorphism i.e. $\displaystyle T^{1}$ is continuous.
Could you please help me in solving this problem.
Regards,
Raed.

A proof from Munkres "Topology" 2nd edition, page 167 is shown in the image below. Note that a metric space is normal (Th 32.2 in P202 of this textbook) and therefore Hausdorff.
http://i56.tinypic.com/9qfy1u.png

Thank you very much for your reply.