Find an infinite collection of subsets of $R$ such that it is closed under countable unions and intersections, but not a $\sigma$- algebra

Question

Find an infinite collection of subsets of $R$ that contain $R$, is closed under the formation of countable unions, and is closed under the formation of countable intersections, but is not a $\sigma$-algebra. Can someone give an example? Thanks.

Answer 1

If you take the set $\{(-n,n); n\in \mathbb N\}\cup \mathbb R$ isn't this closed under countable unions and countable intersections? But it is not closed under compliments so its not a sigma algebra

Answer 2

Consider X, the set of all countable subsets of $\mathbb{R}$. Now, X $\cup \, \mathbb{R}$ is such an infinite collection, it can be seen that this set is closed under both countable unions and countable intersections, but It is not a $\sigma$-algebra because it does not contain complements of its elements (except the pair $\mathbb{R}$-$\emptyset$).