1) Let $\mathcal A$ a set and $\mathcal C=\sigma (\mathcal A)$ the smallest $\sigma -$algebra that contain $\mathcal A$. Does any element of $\mathcal C$ can be written as union an intersection of element of $\mathcal A$ ? I.e. if $B\in \mathcal C$ are there $(A_n^m)_{n,m}\subset \mathcal A$ s.t. $$B=\bigcup_{i=1}^\infty \bigcap_{j=1}^{n_i}A_i^j\tag{*}$$ where $n_i$ can be infinite. Or if it's not exactly like that is there something like that ?
2) Because when $B$ is a Borel set I always see it as union/intersection of open, but may be I'm wrong. So if it's not correct, do you have an exemple of Borel set that can be written as a similar form of $(*)$ ?
For 1) the answer is clearly no. Consider for example $\mathcal{A}=\{1\}\subseteq \mathbb{R}$ then $\mathbb{R}$ is of course in the induced sigma algebra but can not be written in the desired way.