open sets of $\mathbb{N}$

Question

I would like to define a $\mathbb{N}$-valued random variable to define the $\sigma$-algebra generated by this random variable. In my course, we've define only $\sigma$-algebras on borelians of $\mathbb{R}$ ( by the set $\{ X^{-1}(A), A \in B(\mathbb{R}) \}$ for the real case). So what would be borelian of $\mathbb{N}$? Is that just disjoint elements of $\mathbb{N}$ ? Thanks