Definition 4.2.7 (Generated Filter). Let $X$ be a set and $\fB_{0} \subset 2^{X}$ be a filter subbase, then there exists a filter containing $\fB_{0}$.
The smallest filter $\fF(\fB_{0}) \subset 2^{X}$ containing $\fB_{0}$ is the filter generated by $\fB_{0}$, which is given by $\fF(\fB_{0}) = \bracs{E \subset X| \exists F \in \fB: F \subset E}$, where
Proof. For any $\seqf{E_j}, \bracsn{F_j}_{1}^{m} \subset \fB_{0}$,
Thus $\fB$ satisfies (FB1). Since $\bigcap_{j = 1}^{n} E_{j} \ne \emptyset$, $\emptyset \not\in \fB$, and $\fB$ satisfies (FB2).
By Proposition 4.2.3, $\fB$ is a base for the filter $\fF = \bracs{E \subset X| \exists F \in \fB: F \subset E}$.
If $\fF' \supset \fB_{0}$ is a filter, then $\fF \supset \fB$ by (F2), and $\fF' \supset \fF$ by (F1). Thus $\fF$ is the smallest filtetr containing $\fB$.$\square$