let’s say

$mathbf{A}=${all intervals contained in [0,1]} and

$mathbf{B}$={all finite unions of elements of $mathbf{A}$}

proving that $mathbf{B}$ is an algebra is mostly straightforward, except i’m not sure how to prove that $emptyset subset mathbf{B}$. And for that matter, how do I even prove that $emptyset subset mathbf{A}$? Can I just say the ‘interval’ for example $[frac{1}{2},frac{1}{2}]$ it not actually an interval is therefore $emptyset$ ?

To prove that $mathbf{B}$ is *not* a sigma-algebra, do I need to construct a countable union or intersection of subsets of $mathbf{B}$ and show that it’s not contained in $mathbf{B}$? Can I say that $bigcup_{j in mathbf{N}} [frac{1}{2j+1},frac{1}{2j}] notin mathbf{B}$ (where $mathbf{N}$ is the set of natural numbers) and thus $mathbf{B}$ is not closed under countable union and therefore cannot be a sigma-algebra?