homeomorphism between maninifolds

Question

Exist a local homeomorphism between the manifolds with boundary $[0,1) \times [0,1) $ and $\mathbb{R}^{2}_{+}$? I don't think that a local homeomorphism like this can exist..

Answer 1

An initial map from $[0,1)\times [0,1)$ to a quarter circle can be done by taking $(x,y)$ and dividing by the length $\sqrt{x^2+y^2}.$

Now a map from the quarter circle to a semicircle can be defined by the complex map $f(z)=z^2.$ At this point you have arrived at a half-disc, in which the points on the bounding semicircle of radius 1 are not in the half-disc (except that the points of $(-1,1) \times \{0\}$ are in the half-disc). Finally apply the radial map sending $r$ to $r/(1-r)$ to expand the half disk to cover the upper half-plane (including the $x$ axis).

Answer 2

hint $$r\exp(i(\frac{\pi}{2}+\theta)):[0,1)\times[0,1)\rightarrow\text{half disk}$$

I think then you can easily make a half disk into $\mathbb H$.

Attention: $\mathbb R^2_{+}$ is without boundary; $\mathbb H$ is half plane.