If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I don't know the answers to:
Question 1. What is a counterexample when $X$ and $Y$ are non-separable?
Question 2. If $X$ is an uncountable discrete metric space, does
$B(X) \times B(X)$ generate the Borel $\sigma$-algebra on $X \times X$?
Question 3. If $X$ and $Y$ are metric spaces, with $X$ separable, does
$B(X) \times B(Y)$ generate the Borel $\sigma$-algebra on $X \times Y$?
Best Answer
Q1. Discrete spaces with cardinal > c ... then the diagonal is a Borel set, but not in the product sigma-algebra.
This also answers Q2 (no)
but not Q3.