How to write $\sigma(C)$?

Question

Suppose you are given a collection $C$ of subsets of a set $\Omega$. You can always complete this set to a $\sigma$-algebra as follows. Let $F(C)$ denote all $\sigma$-algebra of $\Omega$ that contains C; the $\sigma$-algebra generated by C is

$$\sigma(C) =\bigcap_{G\in F(C)}G$$

Suppose $\Omega =\{1,2,3,4,5,6\}$ and $C=\{\{1,2\},\{2,3\}\}$

Then how can I write $\sigma(C)$?

As I guess, firstly I found its atoms. But I cannot proceed this question

My answer is $\sigma(C)=\{\emptyset, 1, 2, 3, \{1,2\}, \{2,3\}\}$

Answer 1

Yes, firstly find the atoms of $\sigma(C)$. There are FOUR atoms: $\{1\}$, $\{2\}$, $\{3\}$, $\{4,5,6\}$. So, we know that $\sigma(C)$ must have $2^4$ sets. In fact:

$\sigma(C)=\{\emptyset, \{1\}, \{2\}, \{3\}, \{4,5,6\}, \{1,2\}, \{1,3\}, \{1,4,5,6\},\{2,3\}, \{2,4,5,6\}, \{3,4,5,6\}, \{1,2,3\}, \{1,2,4,5,6\}, \{1,3,4,5,6\}, \{2,3,4,5,6\}, \{1,2,3,4,5,6\}\}$.