Borel sigma-algebra of a product of second-countable spaces

Choose at most countable bases $\mathcal{U}$ for $X$ and $\mathcal{V}$ for $Y$ using [F1]. Their product $\mathcal{U}\times \mathcal{V}$ is at most countable by [F2], and its members $U\times V$ form a basis for $X\times Y$ by [F3]. Each such rectangle lies in $\mathcal{B}(X)\otimes \mathcal{B}(Y)$; every open subset of $X\times Y$ is a union of a subfamily of this countable basis, hence a countable union of those rectangles. By [F4], this proves $\mathcal{B}(X\times Y)\subseteq \mathcal{B}(X)\otimes \mathcal{B}(Y)$.

The coordinate projections ${\pi }_X:X\times Y\to X$ and ${\pi }_Y:X\times Y\to Y$ are continuous by the definition of the product topology [F3]. For Borel $A\subseteq X$ and $C\subseteq Y$, $A\times C={\pi }_X^{-1}(A)\cap {\pi }_Y^{-1}(C)$ is Borel in $X\times Y$. Thus every generator rectangle of the product sigma-algebra [F5] belongs to $\mathcal{B}(X\times Y)$, giving the reverse inclusion.