Hatcher, Algebraic Topology, Chapter 0
13. Show that any two deformation retractions and of a space onto a subspace can be joined by a continuous family of deformation retractions , , of onto , where continuity means that the map sending to is continuous.
Proof. Given a topological space and a subspace with two deformation retractions , consider the map for which
for all , .
It’s clear that the map is continuous due to the fact that it’s a straight-line continuous mapping of one continuous map to another . In order to show that the map defines a deformation retraction, note that for arbitrary , is a linear combination of two deformation retractions. In particular, this means that for arbitrary ,
- for all .
Here, not that property 1 above stems from the fact that , property 2 from the fact that , and property 3 from the fact that for all . Hence, as claimed, is a deformation retraction for arbitrary which connects to , whereby the claim is proved.