Homotopy Extension Property involving mapping cylinder Suppose we have a map $f:X\to Y$ and we form the mapping cylinder $M_f$. Hatcher claims that it is obvious that the pair $(M_f, X \cup Y)$ satisfies the homotopy extension property. Equivalently we could find a retraction of $M_f \times I$ to $M_f\times \{0\} \cup (X \cup Y)\times I$. I don't see how we can get this latter result, however.
 A: Neil has given an explicit retraction.  But it may be useful to note that you can obtain results like this from a combination of some "easier" facts:


*

*The pair $(I,\{0,1\})$ has the HEP.

*If $(L,K)$ has the HEP where $K$ and $L$ are locally compact Hausdorf, and if $Z$ is any space, then $(Z\times L, Z\times K)$ has the HEP.

*If $(U,A)$ has the HEP, and $g:A\to B$ is any map, then $(V,B)$ has the HEP, where $V$ is the pushout of $U$ along $g$.
(I'll leave the proofs of these as an exercise; you only need the second one for $(L,K)=(I,\{0,1\})$ anyway.)  Then note that $M_f$ can be obtained from $X\amalg Y$ by gluing it to a copy of $X\times I$ along $X\times \{0,1\}$.
A: I'll assume you want the convention where $M_f$ is $(X\times I)\cup Y$ with $(x,0)$ attached to $f(x)$.  Now $M_f\times I=(X\times I^2)\cup(Y\times I)$ with $(x,0,t)$ attached to $(f(x),t)$.  We want to retract this onto the space 
$$ Q=(M_f\times\{0\})\cup(((X\times\{1\})\cup Y)\times I) $$
Note that $X\times\{0\}\times I$ gets identified with part of $Y\times I$ and so is contained in $Q$.  Thus $Q=(X\times U)\cup(Y\times I)$, where
$$ U=(\{0,1\}\times I)\cup (I\times\{0\}), $$
and again $(x,0,t)$ is attached to $(f(x),t)$.  Now let $r$ be a retraction from $I\times I$ onto $U$, say by radial projection from the point $(1/2,1)$.  We can then fit $1\times r:X\times I^2\to X\times U$ together with the identity map on $Y\times I$ to get the required retraction of $M_f\times I$ onto $Q$.
A: This question is answered by Chapter 7 "Cofibrations", Example 2 on p. 280 of my book `Topology and groupoids' with full proof. In fact it was in the first (1968) edition of this book, published by McGraw Hill. 
Other things in that Chapter are a gluing theorem for homotopy equivalences, the exact sequence of a fibration of groupoids, ....
In other Chapters you will find the Phragmen-Brouwer property, the Jordan Curve Theorem, covering morphisms of groupoids, the fundamental groupoid of an orbit space, ...
See http://groupoids.org.uk/topgpds.html
