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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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{\infy}=C_0 C_1 ...}
you can always contract along like moving contracting the equator along the surface of the earth.