Hatcher, *Algebraic Topology*, Chapter 2, Section 1

**27. Let be a map such that both and the restriction are homotopy equivalences.**

(a) Show that is an isomorphism for all .

**(b) For the case of the inclusion , show that is not a homotopy equivalence of pairs – there is no such that and are homotopic to the identity through maps of pairs. [Observethat a homotopy equivalence of pairs is also a homotopy equivalence for the pairs obtained by replacing and by their closures.]**

*Proof.*