In explaining quotient spaces Martin Crossley (Essential Topology) uses the closed disk in the plane (see attachment showing Crossley's section on Quotient Spaces)

He states on page 78 (see attachment) that is homeomorphic with , by

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

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Crossley then goes on to state that is homeomorphic with 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