Proof That Homotopy Mappings of Homotopic Functions Are Homotopic
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.