For questions about the Baire space, that is the family $\mathbb{N}^\mathbb{N}$ of all sequences of natural numbers with the product topology. For questions about the class of Baire spaces - spaces in which Baire category theorem holds - use (baire-category) tag.
The Baire space, variously denoted $\mathbb{N}^\mathbb{N}$, $\omega^\omega$, or $\mathcal{N}$, is the set of all sequences of natural numbers with the product topology taking $\mathbb{N}$ to be discrete. It is completely metrizable, for example with the metric defined by $d(x,y) = 2^{-n}$ where $n$ is least such that $x(n) \neq y(n)$.
The Baire space is one of the standard spaces in descriptive-set-theory, although the descriptive-set-theory tag should only be used when the questions concern "definability", "uniformization" or other such concerns. For general topological properties (compactness, Lindelöfness, etc.) simply use the general-topology tag.
Note that the name Baire space is also used for spaces in which Baire category theorem holds. This is different thing than the Baire space. For questions about this notion of Baire space, use baire-category tag.
