I have to show that $\displaystyle N^{\mathbb{N}}$ is homeomorphic to $\displaystyle N$, were N stands for Baire space.

The suggested homeomorphism is this:

$\displaystyle \alpha=(f_0, f_1, \ldots) \in N^{\mathbb{N}}$,

$\displaystyle \phi(\alpha)=(f_0(0), f_0(1), f_1(0), \ldots)$

First of all, I don't understand the definition of that homeomorpism, that is, I don't know what comes next after $\displaystyle (f_0(0), f_0(1), f_1(0), \ldots)$

Second of all, I have problems understanding continuity of functions on N, could someone help proving that the above mapping is indeed homeomorphism?

Thank you for every hint.