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

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