Consider a set $E$ and the sigma algebra generated by $C\subset\mathcal{P}(E)$. I would like to know where we can write $C\in\sigma(C)$ or $C\subset\sigma(C)$.
My thoughts were that if $C$ is a set, there is no problem to write $C\in\sigma(C)$ by definition of a sigma algebra generated by a set but the inclusion seems incorrect. Indeed, consider $E=\{1,2,3,4\}$ and
$$ \sigma(\{1\}) = \{ \emptyset, E, \{1\}, \{2,3,4\} \} $$
it's clear that the set $\{1\}$ is in $\sigma(\{1\})$ but $1\not\in\sigma(\{1\})$ so $\{1\}$ is not included in $\sigma(\{1\})$.
However, if $C$ is a class (following the definition provided by Paul Halmos in his book on measure theory), I think that the use of $\in$ is not correct since the set itself could not be in the sigma algebra (but the elements of the set, that are also set, yes). If I consider the previous example with the class $C = \{\{1\}, \{2,4\}\}$, it seems that each element will be found in the sigma algebra generated by this class $C$, but not $C$ itself.
I would like to know if I am wrong or not and have your insights on this please.
Thank you a lot
Thanks to Andrew a mistake leading to a contradiction has been changed.
The term $C$ generates a sigma algebra does only make sense if $C\subseteq\mathcal{P}(E)$. So $C$ is a set of subsets of $E$. And the sigma Algebra generated by $C$ is the smallest sigma algebra containing $C$. So this sigma algebra is again a set of subsets of $E$ with $C\subseteq\sigma(C)$. If $C\subseteq E$ then $\sigma(C)$ does not make sense.