How can we have multiple sigma algebras?

In my textbook, one of the examples shows an $\Omega$ that has two sigma algebras($\mathcal F$ and $\mathcal B$). My only question is, how is it possible for there to be more than one sigma algebra? For example, suppose $\Omega$ is {1,2}. Then the sigma algebra would be {$\Omega$, 1 , 2 , {1,2} , $\emptyset$}. What other ones could exist? Sorry if this seems like a simple question, I'm just having trouble grasping my head around it.

• $\{\emptyset, \Omega\}$ is a $\sigma$-algebra as well. Why would you think that a set must have only one $\sigma$-algebra? – user384138 Jan 27 '17 at 15:56
• I think I'm sort of understanding it now. Would the set {1,2} still be a $\sigma$-algebra as well? – idude Jan 27 '17 at 15:58
• No because a $\sigma$-algebra must have $\emptyset$ and the whole space as elements. – user384138 Jan 27 '17 at 15:59
• Oh ok, I think I get it now. So in the same manner, {$\Omega$, $\emptyset$,{1,2}) would be a sigma algebra too? – idude Jan 27 '17 at 16:02
• $\{1,2\}$ is just $\Omega$ (smh) – user384138 Jan 27 '17 at 16:02