Definition

Given two subsets $U_1,U_2\subset V$, we define the intersection of $U_1$ and $U_2$ to be $$\begin{align*} U_1 \cap U_2 :=\{u\in V|u\in U_1 \text{ and } u\in U_2\} \end{align*}$$ More generally for $U_1,...\subset V$ are all subsets of $V$ then the intersection of $U_i$ is $$\begin{align*} \bigcap_{i=1}^{\infty}U_i:=\{u\in U_i, i=1,2,3,....\} \end{align*}$$

Remark

The intersection $\bigcap\limits_{i=1}^\infty U_i$ is the largest subset of $V$ that is contained in each of $U_i$.