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The Fundamental Group of the Sphere, Radius 1

The space generated by the sphere radius 1 is

Letbe any point ofis homeomorphic toand so is a contractible space. Ifis the antipodal point totaketo be the tangent plane toat

The projection of a pointis a pointdetermined by the intersection of the line throughandwith the planehenceandare homeomorphic.

Ifis a loop inandis any point inthenis homeomorphic to a loop in

Thusis homotopic to the identity function f ofHence any loop inbased onis homotopic to

Henceconsists only of the identity function

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