The intersection of a family $\mathcal{F}$ of sets is the set $\bigcap _{X\in \mathcal{F}}X$ defined by
\[ \bigcap _{X\in \mathcal{F}}X\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ z\in \bigcup _{X\in \mathcal{F}}X\ \middle |\ \text{for each $X\in \mathcal{F}$, we have $z\in X$}\webright\} . \]