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!