Subject: Re: question and job
Date: Fri, 7 Nov 2003 14:31:18 -0500
From: Allen Hatcher <hatcher@math.cornell.edu>
To: Don Davis <dmd1@lehigh.edu>

I hope this answers the question that was being asked by Philippe
Gaucher:

If  i:A ---> X  is a cofibration, then so is (Id,i):YxA-->YxX  for any
space  Y.

Namely:

1) A cofibration is always an inclusion of a subspace.
2) A pair  (X,A)  satisfies the homotopy extension property if and only
if  X x I  retracts onto  (A x I) U (X x {0}).
3) If  X x I retracts onto  (A x I) U (X x {0})  then  X x Y x I
retracts onto  (A x Y x I) U (X x Y x {0})  in the obvious way.

If a reference for these elementary facts is wanted, item 1) is proved
on page 460 of my Algebraic Topology book, and 2) is on page 14.

Allen Hatcher