In the product $X\times Y$ of two uncountable complete separable metric spaces, not every $(s)$-set belongs to the $\sigma$-algebra generated by the products of $(s)$-sets in $X$ with $(s)$-sets in~$Y$. The construction makes use of the fact that the Boolean algebra $(s)/(s_0)$ is complete.
References
J. Morgan, On product bases, Pac. J. Math. 99(1) (1982), 105–126. MR651489 euclid.pjm/1102734137
J. Morgan, On product bases, Pac. J. Math. 99(1) (1982), 105–126. MR651489 euclid.pjm/1102734137
K. Schilling, Some category bases which are equivalent to topologies, Real Anal. Exchange 14(1) (1988), 210–214. MR988366 0678.54021 K. Schilling, Some category bases which are equivalent to topologies, Real Anal. Exchange 14(1) (1988), 210–214. MR988366 0678.54021