# Hatcher 0.27

Hatcher, Algebraic Topology, Chapter 0

27. Given a pair $(X,A)$ and a homotopy equivalence $f:A\to B$, show that the natural map $X\to B\sqcup_f X$ is a homotopy equivalence if $(X,A)$ satisfies the homotopy extension property [Hint: Consider $X\cup M_f$ and use the preceding problem]. An interesting case is when $f$ is a quotient map, hence the map $X\to B\sqcup_f X$ is the quotient map identifying each set $f^{-1}(b)$ to a point. When $B$ is a point, this gives another proof of Proposition 0.17.

Proof.

$\square$