SCHWEITZER 1992 Problem 8

Let $F$ be a set of filters on X so that if $\sigma, \tau \in F$ , $\forall S \in\sigma$ , $\forall T\in\tau$ , we have $S \cap T\neq\emptyset$ , then $\sigma \cap \tau \in F$ . We say that $F$ is compatible with a topology on X when $x \in X$ is a contact point of $A\subset X$ , if and only if , there is $\sigma \in F$ such that $x \in S$ and $S \cap A \neq\emptyset$ for all $S \in\sigma$ .

When is there an $F$ compatible with the topology on X in which finite subsets of X and X are closed ?

contact point is also known as adherent point.