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By Riemann mapping Theorem, we have that there is a comformal mapping from a half plane to a unit disk.

That means, there is a homeomorphism from a half plane to a unit disk.

However, homeomorphism preserves the compactness.

Then, can we conclude from here that a half plane is compact? (Which is a contradiction since a half plane is not closed and bounded.)

There should be some error that I am making in this logic, but I can't find it..

Any comment would be grateful!

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