Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder? Let $ f:X\to Y $ be a map in the pointed category of topological spaces $ Top_* $. And let $ U:Top_*\to Top $ be the "forgetful" functor (which "forgets" the basepoint). We can look at the reduced mapping cylinder $ M_f $ and at the unreduced mapping cylinder $ M_{U(f)} $ in the category $ Top $.
Until yesterday, I thought $ M_f $ has the same homotopy type of $ M_{U(f)} $. But, in May's Concise Course, he says "If $X, Y$ are well-pointed, then $ M_f $ has the same homotopy type of $ M_{U(f)} $."
Is this hypothesis necessary?
If it is truly necessary, I want to know where I'm wrong.
I thought this: Since $ M_{U(f)} $ is a pushout of a trivial cofibration, we have that $ Y\to M_{U(f)} $ is a trivial cofibration.
The same way (or only using a explicit homotopy), we have that $ Y\to M_f $ is a homotopy equivalence. So we have that $ M_{U(f)}\equiv Y\equiv M_f $.
I know that the first statement is right. If there is something wrong, it is in the second statement. 
I believed that we can factor any function in $ Top _* $ in the same way as in $ Top $, id est,
$ f= R\circ j $, where $ j: X\to M_f $ is a cofibration and $ R: M_f\to Y $ is a strong retract. Is it wrong?
 A: I don't trust anything with degenerate basepoints, but I do think you are right about the homotopy type via the parenthetical  "(or only using an explicit homotopy)''.  However, I was working in compactly generated spaces, denoted $\mathcal{U}$, including the weak Hausdorff property. I don't think you can prove that $Mf$, constructed as usual, is in $\mathcal{U}$ without assuming nondegenerate basepoints.   Also, without nondegenerate base points, you cannot be sure that $j$ is a cofibration (at least not in the unbased sense; see p. 56 of Concise).  In More Concise, Ponto and I take $\mathcal{T}$ to mean nondegenerately based spaces and $\mathcal{U}_*$ to mean based spaces in $\mathcal{U}$.  This is justified model theoretically by noting that $\mathcal{T}$ is the full subcategory of $h$-cofibrant objects in $\mathcal{U}_{*}$ (that is, cofibrant in the based Hurewicz model structure on $\mathcal{U}_{*}$).  Incidentally, one reason to start work in $\mathcal{U}_{*}$ and not
$Top_{\ast}$ is that the smash product in $Top_{\ast}$ is not associative (a published source for a very old counterexample is Parametrized homotopy theory, by Sigurdsson and myself). 
A: I'm lousy at point-set topology, always was, but I don't see that $X$ is closed in $X\wedge I_+$.   This is the sort of question a working algebraic topologist does not want to think about.  Cofibrant
approximation in the $h$-model structure on $\mathcal{U}_{*}$ takes $X$ to the whiskered space $X\vee I$ with new basepoint at $1$ if the basepoint of $I$ is taken to be $0$.  If $X$ has a nasty degenerate basepoint, take the nasty thing away: I do not want to think about it.  
