**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.