existence of space and deformation retraction

Question

Consider a homotopy equivalence between two non-empty topological spaces $M$ and $N$.

Prove there exist a topological space P and inclusions $i : M → P,\ j : N → P$ such that $i(M ) ⊂ P, \ j(N ) ⊂ P$ are deformation retraction.

I think of the obvious direct product $P=M\times N$. Would it be a good start to tackle this question?

Thank you!

Answer 1

Try instead $M_f$, the mapping cyllinder for a homotopy equivalence $f:M \to N$. This is $(M \times I) \coprod N/\sim$ where $(m,1) \sim f(m)$.

Note that this retracts onto $N$ by just going along $(x,t)$. How about $M$? See Hatcher Chapter 0 for how this all goes.