Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces. Take $$\mathcal{B}(X_1)\times \mathcal{B}(X_2)=\sigma(\{A_1\times A_2 | A_1\in \mathcal{B}(X_1), A_2\in \mathcal{B}(X_2) \})$$ Where $\sigma(A)$ denotes the smallest $\sigma$-algebra generated by $A$, and $\mathcal{B}$ denotes Borel $\sigma$-algebra.
Define the following metric on the product space: $$d((x_1,x_2),(y_1,y_2))=\sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$$
Let $X=X_1\times X_2$, and $$\mathcal{B}(X)=\sigma(\{ \textrm{All open sets in } (X,d)\})$$
If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces then $\mathcal{B}(X)=\mathcal{B}(X_1)\times \mathcal{B}(X_2)$.
Is it possible to show by counter-example that separability is necessary in that assertion?
