$\newcommand{\B}{\mathcal{B}}$I read on wikipedia that if $X$ and $Y$ are second-countable Hausdorff spaces, then the identity is an isomorphism $\B(X)×\B(Y)≅\B(X×Y)$. I don't understand the requirement of $X$ and $Y$ being Hausdorff. I also read that requirement on MathOverflow (where only one space is asked to be second-countable). Is there a flaw in the proof below?
Let $X$ and $Y$ be topological spaces. Let $(U_i)_{i∈I}$ be a countable basis of the open sets of $X$. The identity $\B(X×Y)→\B(X)×\B(Y)$ is always measurable by functoriality of $\B$. Let $O⊆X×Y$ be open. The goal is to prove that is it measurable in $\B(X)×\B(Y)$.
Each such $O$ can be written as a union of products of the form $U_i×V$ with $i∈I$ and $V⊆Y$ open. In that union, for each $i∈I$, we can group the products $U_i×V$ in a single product of that form. So $O$ can be written as the countable union $⋃_{i∈I} U_i×V_i$ with $V_i⊆O$ open. But each $U_i×V_i$ is measurable in $\B(X)×\B(Y)$, so $O$ is.