I am trying to prove closed upper half-sphere $H_2^+/{\sim}$ is homeomorphic to $B^2/{\sim'}$ where $\sim$ identifies antipodes on the equator of $H_2^+$ and $\sim'$ identifies antipodes on the boundary of $B^2$.
I know that $B^2$ is homeomorphic to the the closed upper half-sphere $H_2^+$ through $(x,y) \mapsto \Big(x,y,\sqrt{1−x^2-y^2}\Big)$
Also if points are antipodes on the boundary of $B^2$, then their image is antipodes of the equator of $H_2^+$ so this quotient should preserve the homeomorphism. However, I am trying to show this rigorously (at least more rigorously).
Thanks