Products of Product Spaces

Let $\mathbb{N}$ be Natural numbers set with the discrete topology $\tau_{D}$ induced by the discrete metric. Then $\mathbb{N}$ copies of $(\mathbb{N},\tau_{D})$ gives us the Baire space $(\mathbb{N}^{\mathbb{N}},\tau)$ where $\tau$ is the product topology. Is there something like $\mathbb{N}$ copies of the Baire spaces ? If so, what we can say for it’s topology and structure ? Or it’s also a set with product topology? Any help will highly be appreciated, thank you very much.

Yes, this is again a product space of $|\Bbb N \times \Bbb N|$ many (so again $\Bbb N$) many) copies of the discrete space, so in fact homeomorphic to Baire space again.

This is a special case of the transitive law of initial topologies, plus cardinality considerations.

Any finite or countable product of Baire space is again Baire space. Ditto for the Cantor set $\{0,1\}^{\Bbb N}$ and the Hilbert cube $[0,1]^{\Bbb N}$ etc. It's a common thing.