Filterbasis eines Mengenfilters

Let \(X\) be a set and \(\mathcal{S} \subseteq \mathcal{P}(X)\) be a system of sets such that a finite intersection \(\bigcap_{i =1}^n S_i \neq \emptyset\) is nonempty. Then \(\mathcal{S}\) is a filterbasis for the setfilter

\begin{align*} \mathcal{F} \coloneqq \{A \subseteq X \vert \exists S_1,...,S_n : \bigcap_{i=1}^n S_i \subseteq A\} \end{align*}