Proof That The Homotopy Property is Reflexive
Theorem
If
and
are continuous functions and
is homotopic to
- written
thenis homotopic to
Proof
Since
there is a homotopy
with
Define
by
Then
and
Since
is continuous, so is
and
Theorem
Ifandare continuous functions andis homotopic to- writtenthenis homotopic to
Proof
Sincethere is a homotopywith
Defineby
Thenand
Sinceis continuous, so isand