Let
and define
Claim
is contractible
Proof
Suppose
then
Define
by
by
by
and so on ...
applying the homotopy
in the time interval
,
in the interval
,
in
etc...
we get the desired contraction to the point
.
A different way to see this is via the cellular structure of
. If Failed to parse (unknown function "\infy"): {\displaystyle S^{\infy}=C_0C_1...}
you can always contract
along
like moving contracting the equator along the surface of the earth.