If X is a CW complex, and 
is a subcomplex,
we want to prove that has the homotopy extension
property.
This is the same as proving that 
is as deformation retract of 
.
The first remark is that since I=[0,1] is a finite CW complex,
the product topology on
agrees with the CW topology on , given by
the cell structure of . So in order to prove that
a homotopy 
is continuous, it suffices to prove that it is continuous on
each skeleton of 
.
The second remark is that we can construct homotopies on each skeleton
such that 
is the identity, and 
is
a retraction
We define a new homotopy
inductively in the following way:
Assume that the homotopy is defined on 
,
in such a way that it 
is the identity for 
.
We define 
on 
as follows:
This is welldefined, since for ,
and continuous on each , so
by the definition of the topology of the CW complex,
it is continuous on .