If a Borel $\sigma$-algebra is genertated by a collection, then the collection is closed under intersections?

Question

The question is:

If $\sigma\left(\tau\right)$ is a Borel $\sigma$-algebra such that is genertated by a collection $\mathcal{B}\subseteq\sigma\left(\tau\right)$, then $\mathcal{B}$ is closed under intersections?

I try to find a contraexmple but I have not succeeded.

Answer 1

The discrete Borel $\sigma$-algebra on $\mathbb{N}$ is generated by $\{ \{i, i+1\} : i \in \mathbb{N}\}$, which is clearly not closed under finite intersections.

Or, if you only need a counter-example in the infinite intersection case, take the usual Borel $\sigma$-algebra on $\mathbb{R}$ with generating collection the open intervals, and note that $\bigcap_n (0-\frac{1}{n}, 1+\frac{1}{n}) = [0,1]$.