Definition

Given two subsets $U_1, U_2 \subset V$, we define the union of $U_1$ and $U_2$ to be $$\begin{align*} U_1\cup U_2:=\{u\in V|u\in U_1 \text{ or } u\in U_2\} \end{align*}$$ More generally, for $U_1,....\subset V$ the union of all $U_i$ is $$\begin{align*} \bigcup_{i=1}^\infty U_i:=\{u\in V|u\in U_i, \text{ for some } i=1,2,3,\} \end{align*}$$ ## Remark The union $\bigcup\limits_{i=1}^\infty U_i$ is the smallest subset of $V$ that contains each of $U_i$.