Hatcher, Algebraic Topology, Chapter 0
12. Show that a homotopy equivalence induces a bijection between the set of path-components of and the set of path-components of , and that restricts to a homotopy equivalence from each path-component of to the corresponding path component of . Prove also the corresponding statements with components instead of path-components. Deduce that if the components of a space coincide with its path-components, then the same holds for any space homotopy equivalent to.
Proof. Let be topological spaces and let be a homotopy equivalence with homotopy inverse . For notational convenience, let denote the sets of path-components of and define a map as follows: For every , let denote the path-component of containing 1 and define to be the map2 for which . It suffices to show that this map is well-defined and that it indeed is a bijection as claimed.
To show well-definedness, suppose that in , i.e. that and lie in the same path component of . In particular, there is a continuous map for which and . Consider the map . This map is continuous as it’s the composition of continuous maps, and it sends to , meaning that and lie in the same path component in . Thus, and so the map is well-defined.
Finally, to show that is a bijection, it suffices to show that has a well-defined inverse . Define for all . To show that these functions really are inverses, note that and that, similarly, . By definition of homotopy inverse, there are homotopies and so that and . In particular, these homotopies defined “paths” connecting the maps to and to , whereby it follows that and are in the same path components as and , respectively. Hence, and for all and so it follows that are inverses.
1. The notation for the path-component of containing is suggestive of the presence of equivalence classes / quotient maps. Indeed, this stems from the fact that path-connectedness is an equivalence relation – a fact that’s well-known from point-set topology.
2. After floundering for a few hours trying to make my poorly-defined original map work, I stumbled upon this document online: http://www.stanford.edu/class/math215b/Sol1.pdf. Therefore, the idea for this particular definition belongs to him/her/them.