In his text book on descriptive set theory, Proposition 25.2, Kechris implicitely uses the fact that $(\omega \times \omega)^\omega$ and $\omega^\omega \times \omega^\omega$ are homeomorphic, where $\omega$ denotes the set of non-negative integers and carries the discrete topology.
Actually, the following is not hard to see (if I did not make a mistake):
Let $X_1, \ldots , X_n$ be discrete spaces and let $Y$ be a set. Then the product spaces $X_1^Y \times \ldots \times X_n^Y$ and $(X_1 \times \ldots \times X_n)^Y$ are homeomorphic.
I would need a reference for this fact.
Thank you in advance for any help!