Massey Exercise 2.7.5

Theorem: Let be a topological space, and a continuous map. Let be such that:

  • For any ,
  • and for any

Where and are defined as in 7.4. Then, for any , .

Proof: Because is continuous, we have . Now, consider that , by the second property. Then, , by the first property. That is, , and a symmetric argument obtains . Then, .

By the last exercise (7.4), and commute, so for any , we have

So, is an abelian group.