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> Find a homeomorphism from the whole plane to the open upper half plane.
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So I need to find a bicontinuous map $f:\mathbb{R^2} \to \mathbb{R} \times[0, \infty)$. If I consider $f(x,y)=(x,e^y)$ then certainly $f$ is continuous since this is a $C^\infty$ function? Both component functions also have a continuous inverse, but neither one is defined at $(x,y)=(0,0)$? Is this a valid candidate or am I just mistaken here?