Jtdylktuy

From what I understood, a ring of sets is a non-empty class of sets that is closed under symmetric difference of any pair of sets of the class and under intersection of any pair of sets of the class.

Why isn't it a set of Boolean algebra and how do you show it to be a ring of sets?
I have just begun Introduction to Topology and Modern Analysis for context.

A ring of sets is defined in Simmons to be a non-empty class $mathcal{A}$ such that for $A,B in mathcal{A}$:



(1) $A Delta B in mathcal{A}$, where $Delta$ stands for the symmetric difference.


(2) $A cap B in mathcal{A}$.



Note that this implies $emptyset in mathcal{A}$. For this proof, you need to show the following. If $B_1, B_2 in mathcal{A}$, where $mathcal{A}$ is the class of all finite unions of closed-open intervals on the real line, then $B_1 Delta B_2 in mathcal{A}$ and $B_1 cap B_2 in mathcal{A}$.



To show that $mathcal{A}$ is not a Boolean algebra of sets, consider whether $B_1'$ is in $mathcal{A}$, where $B_1'$ stands for the complement of $B_1$. To help you get on your way, maybe define $B_1 = [a,b)$ and $B_2 = [c,d)$ where $a,b,c,d in mathbb{R}$. Then consider whether these two closed-open intervals are overlapping, containing one or the other, or disjoint.