Is there a perfect Polish topology on the Baire space that is strictly finer than the usual topology?

Question

The question is: is there a perfect Polish topology on the Baire space $\mathbb{N} ^ \mathbb{N}$ that is strictly finer than the usual topology on $\mathbb{N} ^ \mathbb{N}$? The usual topology is the product topology on $\mathbb{N} ^ \mathbb{N}$ with the discrete topology on $\mathbb{N}$.

Answer 1

Yes, this is a corollary from Lemma 13.2 from “Classical Descriptive Set Theory” by Alexander Kechris.