Consider a $\displaystyle \sigma $-algebra $\displaystyle \mathcal{F} $ on $\displaystyle \mathbb{R} $ containing all intervals $\displaystyle (-\infty, x], x \in \mathbb{R} $. Show that it must contain the sets of the form $\displaystyle (a,b], (a,b) $ and $\displaystyle \lbrace a \rbrace $ $\displaystyle ( $ where $\displaystyle a < b), $ and the countable unions thereof.

In class my lecturer said that I should keep de Morgan's laws in mind, as well as the fact that $\displaystyle \mathcal{F} $ is closed under complements and under countably infinite solutions.