The union of the family $\webleft\{ A_{i}\webright\} _{i\in I}$ is the set $\bigcup _{i\in I}A_{i}$ defined by
\[ \bigcup _{i\in I}A_{i} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in F\ \middle |\ \text{there exists some $i\in I$ such that $x\in A_{i}$}\webright\} , \]