Is my understanding of product sigma algebra (or topology) correct?

For the comments: I retract my error for the definition of measurability. Sorry.

For the two things generating the same sigma algebra (or topology, which is similar):

We use $\langle - \rangle$ to denote the smallest sigma algebra containing the thing in the middle. We want to show that $$(1) \hspace{5mm}\langle \prod_{i} B_i \rangle$$ where $B_i \in \mathcal{B}_i$, and $B_i = E_i$ for all but finitely many $i$s, is the same as $$(2) \hspace{5mm}\langle \prod_{i} B_i \rangle$$ where $B_i \in \mathcal{B}_i$, and $B_i = E_i$ for all but one $i$.

It is clear that $(2) \subset (1)$, since the generating collection in (2) is a subset of that of (1).

On the other hand, $(2)$ contains $\prod_{i} B_i$ where $B_i \in \mathcal{B}_i$, and $B_i = E_i$ for all but finitely many $i$s, since it is the (finite) intersection of the generators. For example, $$B_1 \times B_2 = (B_1 \times E_2) \cap (E_1 \times B_2)$$ So $(2) \supset (1)$. Therefore $(2) = (1)$.