I hope someone will read this long question to the end, I'm getting really desperate.

Here's what I have to prove:
Let $\displaystyle \psi: \mathbb{N}^d \times B^k \rightarrow B$ be defined as follows:
$\displaystyle \alpha=(n_1, \ldots, n_d, f_1, \ldots, f_k)$
$\displaystyle \psi(\alpha)=(n_1, \ldots, n_d, f_1(0), f_2(0), \ldots, f_k(0), f_1(1), \ldots, f_k(1), \ldots)$,
where B is the Polish set $\displaystyle B=\mathbb{N}^{\mathbb{N}}$ with product topology and every $\displaystyle \mathbb{N}$ has discrete topology.
Show that $\displaystyle \psi$ is a homeomorphism.

Injectivity and surjectivity is easy, continuity isn't.

Here are the facts:
$\displaystyle f:B \rightarrow B$ is continuous if $\displaystyle \forall x \in B$ and $\displaystyle \forall \sigma \subset f(x)$ $\displaystyle \exists \tau \subset x$ such that $\displaystyle \tau \subset y \Rightarrow \sigma \subset f(y)$.

$\displaystyle f:X\rightarrow \Pi Y_i$ is continuous if and only if $\displaystyle \pi_i \circ f$ is continuous, for all i.

Here's what I got, after huge online help:
Let $\displaystyle \phi=\psi^{-1}$ be the inverse.
$\displaystyle \phi: B \rightarrow \mathbb{N}^d \times B^k$ is continuous if $\displaystyle p \circ \phi$ and $\displaystyle q \circ \phi$ are continuous, where
$\displaystyle p: \mathbb{N}^d \times B^k \rightarrow \mathbb{N}^d$ takes $\displaystyle (x, y) \in \mathbb{N}^d \times B^k $ to $\displaystyle x\in \mathbb{N}^d$.
$\displaystyle q: \mathbb{N}^d \times B^k \rightarrow B^k$, akes $\displaystyle (x, y) \in \mathbb{N}^d \times B^k $ to $\displaystyle y\in B^k$.

Since $\displaystyle \mathbb{N}^d$ and $\displaystyle B^k$ are also products, we can say that $\displaystyle \phi$ is continuous if $\displaystyle p_i \circ \phi$, $\displaystyle i=1, \ldots, d$ and $\displaystyle q_j \circ \phi$, $\displaystyle j=1,\ldots, k $ are continuous.
$\displaystyle p_i$ sends $\displaystyle (x, y)$ to the i-th coordinate of the $\displaystyle \mathbb{N}^d$part, and $\displaystyle q_j$ sends $\displaystyle (x, y)$ to the j-th coordinate of the $\displaystyle B^k$ part.

Now it's easy to prove that the inverse is continuous.

The question is: what do I do about $\displaystyle \psi$?

Please advise.
Thank you!