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.