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!   

if you agree that the borel $\sigma$-algebra isn't neccessarily all the subsets of the topological space, then you might also agree that $P(X)$ (or $2^X$ in a different notation, the power set) is a larger $\sigma$-algebra. Larger means more sets in the $\sigma$-algebra.
Also note that the generated $\sigma$-algebra by the open sets, which is the borel $\sigma$-algebra, is the intersection of all $\sigma$-algebras containing the open sets - which is smaller than any other $\sigma$-algebra by the sense of being a subset.
