The space generated by the sphere radius 1 is
Letbe 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 pointis a point
determined by the intersection of the line through
and
with the plane
hence
and
are homeomorphic.
Ifis a loop in
and
is any point in
then
is homeomorphic to a loop in
Thusis homotopic to the identity function f of
Hence any loop in
based on
is homotopic to
Henceconsists only of the identity function