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!   

Lets consider $\Omega=\{1,2,3,4\}$
$\sigma(\{1\})$ is the smallest $\sigma$-algebra which contains $1$.
So we must take any other elements of $P(\Omega)$ such that the conditions for being a $\sigma$-algebra are fulfilled.
It does clearly contains $1$. Also $1^C=\{2,3,4\}$. And $\Omega,\emptyset$.
So we have $\sigma(\{1\})=\{\Omega,\emptyset,\{1\},\{2,3,4\} \}$. Indeed this is a $\sigma$-algebra. Also we got this set for taking as much elements of the power until our conditions are fulfilled for the "first time".
An example for a $\sigma$-algebra with is not the smallest one but is containing $\{1\}$ is:
B:=$\{\Omega,\emptyset,\{1\},\{2\},\{1,2\},\{3,4\},\{2,3,4\},\{1,3,4\} \}$.
Clearly $\sigma(\{1\})\subset B$
