I'm having some problems finding the topology for the Baire space \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 U_{n_1, \ldots, n_m} and the sets N_{\sigma} the same, for \sigma=n_1, \ldots, n_m?

Thank you.