Let $C_1$ and $C_2$ are two collections of subsets of the set $\Omega$. We want to show that if $C_2$ $\subset$ $\sigma$[$C_1$] and $C_1$ $\subset$ $\sigma$[$C_2$], then $\sigma$[$C_1$]=$\sigma$[$C_2$] where $\sigma$[$C$] is defined to be the sigma-algebra generated by a collection $C$ of subsets of $\Omega$ (or the smallest sigma-algebra).
I'm having trouble characterizing the smallest sigma-algebra of sets in $C_1$ and $C_2$. Would the smallest sigma-algebra of $C_1$ be {$\emptyset$, $\Omega$, $C_1$, $\Omega$ $\not$ $C_1$} or would we have to characterize them in terms of the sets in $C_1$ which is the countable subcollection of sets in $\Omega$?