I'm having some problems finding the topology for the Baire space $\displaystyle \mathbb{N}^{\mathbb{N}}$.

This is what I've found in one book:
and this is what I've found in another one:

My question is, are these two topologies the same?
That is, are the sets $\displaystyle U_{n_1, \ldots, n_m}$ and the sets $\displaystyle N_{\sigma}$ the same, for $\displaystyle \sigma=n_1, \ldots, n_m$?

Thank you.