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

Question

The question is:

Let $\left(M,\sigma\left(\tau\right),\mu\right)$ a measurable space with $\mu$ a probability.

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 unions?

If the elements of $\mathcal{B}$ are open, change the situation?

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

Answer 1

If $M=\{1,2\}$ endowed with the discrete topology, the collection $\mathcal B:=\{\{1\} ,\{2\}\} $ generates the Borel $\sigma$-algebra and is not closed by unions.

Answer 2

What if the collection is the set of all bounded open intervals on the real line?