Intersection of Events is Event

Theorem
Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

The event space $\Sigma$ of $\mathcal E$ has the property that:
 * $A, B \in \Sigma \implies A \cap B \in \Sigma$

That is, the intersection of two events is also an event in the event space.

Also see

 * Elementary Properties of Event Space