$\cap$-Stable Generator for Canonical Product $\sigma$-Algebra

Question

Let $D$ be a (possibly uncountable) topological space with Borel $\sigma$-algebra $\mathcal{B}(D)$.

Let the space of functions from $D$ to $\mathbb{R}$ $$ \mathbb{R}^{D}: = \{f \mid f : D \to \mathbb{R} \} $$ be equipped by the $\sigma$-algebra $\mathcal{A}(\mathbb{R}^{D})$, which is generated by all the projection maps $$j_{\pi} : \mathbb{R}^D \rightarrow \mathbb{R}, \quad j_\pi (f) = f(\pi)$$ for $\pi \in D$.

What would be a $\cap$-stable generator for $\mathcal{A}(\mathbb{R}^{D})$ ?

Answer 1

I suppoase by $\cap$ stable you mean stable under finite intersections. Sets of the form $\{f\in \mathbb R^{D}: f(d_1)\in A_1,f(d_2)\in A_2,...,f(d_n)\in A_n\}$ where $n$ is a positive integer, $A_i$'s are Borel sets in $\mathbb R$ and $d_i$'s belong to $D$ is one such useful generating class.