The Fundamental Group of the Sphere, Radius 1
The space generated by the sphere radius 1 is
Let
be any point of
is homeomorphic to
and so is a contractible space. If
is the antipodal point to
take
to be the tangent plane to
at
The projection of a point
is a point
determined by the intersection of the line through
andwith the planehenceandare homeomorphic.
If
is a loop inandis any point in
thenis homeomorphic to a loop in
Thusis homotopic to the identity function f of
Hence any loop inbased on
is homotopic to
Hence
consists only of the identity function
