In explaining quotient spaces Martin Crossley (Essential Topology) uses the closed disk $\displaystyle D^2 $ in the plane $\displaystyle \mathbb{R}^2 $ (see attachment showing Crossley's section on Quotient Spaces)

He states on page 78 (see attachment) that $\displaystyle D^2 - \partial D^2 $ is homeomorphic with $\displaystyle \mathbb{R}^2 $ , by

$\displaystyle ( r cos ( \theta ), r sin(\theta) ) \longleftrightarrow ( tan ( \frac {\pi r }{2} ) cos ( \theta ) , tan ( \frac {\pi r }{2} ) sin ( \theta ) ) $

Can anyone help me show explicitly and formally that this is a homeomorphism?

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Crossley then goes on to state that $\displaystyle \mathbb{R}^2 $ is homeomorphic with $\displaystyle S^2 - \{ (0,0,1) \} $ by stereographic projection. (see attachement page 78)

I also need help to explicitly and formally prove that this is a homeomorphism.

Be really grateful for help on either one or both of these problems.

Peter