The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra

In the setting of The Real Numbers: Standing Notation and Background, let $(Y,d_{Y})$ and $(Z,d_{Z})$ be separable metric spaces, let $Y\times Z$ carry the product metric $d_{Y\times Z}$, a metric by claim 1 of The Product Metric is a Metric, and write $\mathcal{B}(Y)$, $\mathcal{B}(Z)$ and $\mathcal{B}(Y\times Z)$ for the Borel $\sigma$-algebras of $(Y,d_{Y})$, $(Z,d_{Z})$ and $(Y\times Z,d_{Y\times Z})$.

¶ The Borel $\sigma$-algebra of the product is the product $\sigma$-algebra:

$$\mathcal{B}(Y\times Z)=\mathcal{B}(Y)\otimes\mathcal{B}(Z).$$