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