Let $(X,\mathcal O)$ be a topological space. If $B$ is a non-empty set of subsets $O\subseteq X$ that does not contain the empty set $\emptyset,$ then the set $F$ of all subsets of $X$ that contain some element of $B$ is a filter if and only if the intersection of any two sets of $B$ contains a set in $B.$
In this case, the set $B$ is called a filter base of $F$ and $F$ is said to be generated by $B.$
Proofs: 1