> [!NOTE] Definition > Let $\mathcal{F}$ be an [[Event Space|event space]] on $\Omega$. Then $\mathcal{F}$ is closed under [[Countable Set|countable]] [[Set Intersection|intersection]] (an intersection of a countably infinite set of sets). **Proof**: Let $A_{1},A_{2}\dots \mathcal \in \; \mathcal{F}.$ By [[De Morgan's Laws for Union]], we have $\bigcap_{j=1}^{\infty} A_{j} = \left( \bigcup_{j=1}^{\infty} A_{j}^{c} \right)^{c}.$Since $\mathcal{F}$ is closed under compliments, $A_{j}^{c}\in \mathcal{F}$, for all $j\in\mathbb{N}^{+}.$ Since $\mathcal{F}$ is a closed under countable unions $\bigcup_{j=1}^{\infty} A_{j}^{c}\in\mathcal{F}$Taking compliments again gives $\bigcap_{j=1}^{\infty} A_{j} \in \mathcal{F}.$