Example of measurable space generating by a collectión $\mathcal{B}$ such that $\mathcal{B}$ is not colsed under intersections.

Question

There is a $\left(M,\sigma\left(\tau\right)\right)$ meausrable space with $M$ no discrete (**$\sigma\left(\tau\right)$ a Borel $\sigma$-algebra where $\tau$ is a topology in $M$)** such that $\sigma\left(\tau\right)$ is generating by a collection $\mathcal{B}\subset \sigma\left(\tau\right)$, but $\mathcal{B}$ is not closed under intersections of the any two elements?

Answer 1

On $\mathbb {R},$ consider the collection of open intervals together with the set $\{0,1\}.$