Theorem
If
are homotopic mappings taking values in the space
and
are defined on and homotopic, then the mappings
and
are homotopic.
Proof
Let
so that
is homotopic to
A continuous function
for all
exists such that
Let
so that
is homotopic to
A continuous function
for allexists such that
Define
Then
and
are continuous, nH is continuous so
is a homotopy andandare homotopic.