# Homeomorphism

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• March 23rd 2011, 05:44 AM
raed
Homeomorphism
Dear Colleagues,

Let $X$ and $Y$ be metric spaces, $X$ compact, and $T:X\longrightarrow Y$ bijective and continuous. Show that $T$ is a homeomorphism i.e. $T^{-1}$ is continuous.

Could you please help me in solving this problem.

Regards,

Raed.
• March 23rd 2011, 07:07 AM
zzzhhh
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
• March 23rd 2011, 11:49 AM
raed
Thank you very much for your reply.