## base for the topology of baire space N^N

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.