Borel $\sigma$-algebra definition question So I am studying measure theory and I have found myself struggling to fully understand the concept of the Borel $\sigma$-algebra in depth. We know that the Borel $\sigma$-algebra is the smallest $\sigma$-algebra containing all open sets. The part that I cannot clearly grasp is the word smallest. The way we can generate a Borel $\sigma$-algebra is take all the open sets and take all possible set operations between them. Won't this always produce a unique $\sigma$-algebra of sets? How can a $\sigma$-algebra be larger? Can anyone provide an intuitive example? Thank you in advance!   

Say we are doing this on the real line $\mathbb R$.  Let $\mathcal G$ be the collection of all open sets.  We are interested in $\sigma$-algebras $\mathcal F$ such that $\mathcal F \supseteq \mathcal G$.  There may be many such $\sigma$-algebras.  For example, the power set $\mathcal P$, consisting of all subsets of $\mathbb R$ is one.  But that is the largest one, we want the smallest one.  
As a comment noted, this largest and smallest are in the sense of set inclusion.  The Borel $\sigma$-algebra $\mathcal B$ satisfies $\mathcal B \subseteq \mathcal F$ for any $\sigma$-algebra $\mathcal F$ such that $\mathcal F \supseteq \mathcal G$.  That is the sense in which $\mathcal B$ is smallest.
