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Filter (set theory)

{title2=$\forall F,G\in\mathcal F\ \exists H\in\mathcal F\ (H\subseteq F\cap G)$}

A <filter base> is a nonempty family of nonempty <subsets> directed under reverse inclusion. Its upward closure is a <filter>[proper filter]. Every <finite intersection> contains a base member, so it has the <finite intersection property>.

Filter base

Connected intersection of a closed filter base

Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2008 / A / 4 / a / Solution

Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / A / 3 / i / Solution

Filter base ( $\forall F, G \in F \exists H \in F (H \subseteq F \cap G)$ )

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Filter base (∀F,G∈F ∃H∈F (H⊆F∩G))

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Filter base ( $\forall F, G \in F \exists H \in F (H \subseteq F \cap G)$ )

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