Wikipedia's definition of Family of sets:
In set theory, a collection $F$ of subsets of a given set $S$ is called a family of subsets of $S$, or a family of sets over $S$.
So suppose $Ω$ is a set, $\mathcal{C}$ is a family set of $Ω$ and $\sigma(\mathcal{C})$ denotes the $\sigma$-algebra generated by $\mathcal{C}$. So by definition of generated $\sigma$-algebra, $\mathcal{C}$ must be a subset of $\sigma(\mathcal{C})$.
But I'm not pretty sure of the relationship between $Ω$ and $\sigma(\mathcal{C})$. I think $Ω$ may not be an element of $\sigma(\mathcal{C})$.
Cite Bungo's comment,
" $\Omega$ must be an element of $σ(C)$, by definition. Review the axioms defining a σ-algebra (let's call it $\Sigma$ in general) over $\Omega$: (1) $\Omega \in \Sigma$; (2) if $A \in Σ$ then $A^c \in \Sigma$; (3) if $(A_n)_{n=1}^{\infty}$ is a sequence of sets where each $A_n \in \Sigma$, then $\cup_{n=1}^{\infty}A_n \in \Sigma$ ".