The union of $\mathcal{U}$ is the set $\bigcup _{U\in \mathcal{U}}U$ defined by
\[ \bigcup _{U\in \mathcal{U}}U\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $U\in \mathcal{U}$}\\ & \text{such that we have $x\in U$}\end{aligned} \webright\} . \]
