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set of filters compatible with topology

Source: miklos schweitzer 1992 q8

October 25, 2021
topologyfiltered algebraabstract algebra

Problem Statement

Let FF be a set of filters on X so that if σ,τ∈F \sigma, \tau \in F , ∀S∈σ\forall S \in\sigma , ∀T∈τ\forall T\in\tau , we have S∩T≠∅S \cap T\neq\emptyset , then σ∩τ∈F\sigma \cap \tau \in F. We say that FF is compatible with a topology on X when x∈Xx \in X is a contact point of A⊂XA\subset X , if and only if , there is σ∈F\sigma \in F such that x∈Sx \in S and S∩A≠∅S \cap A \neq\emptyset for all S∈σS \in\sigma .
When is there an FF compatible with the topology on X in which finite subsets of X and X are closed ?
contact point is also known as adherent point.
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