I read that a topological space $(X,d)$ is a Baire space if for every sequence $\{X_n\}$ of open dense subsets of $X$, the set $\bigcap_{n=1}^{\infty}X_n$ is also dense in $X$.
Since every complete metrizable space is a Baire space, and since $\mathbb{N}$ is a Polish space (which is complete metrizable by definition), it follows that $\mathbb{N}$ is also a Baire space.
But what if I take $X_n = \mathbb{N} / \{x_n\}$ as a sequence of sets. I think (not sure) that each of these sets is open and dense in $\mathbb{N}$, but we have $\bigcap_{n=1}^{\infty}X_n=\emptyset$, which is not dense in $\mathbb{N}$.
Or is it that there's something wrong with my understanding that $X_n$ should be dense - given that the only dense subset of a discrete space is the space itself.
$\mathbb N$ has discrete topology. Every set is open and closed. In particular your sets $X_n$ are closed proper subsets. They are not dense.