 Solution

ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2009/a/3/i/solution

X

is a nonempty family

F

of nonempty subsets such that for every

F, G \in F

there is

H \in F

with

H \subseteq F \cap G

. Repeated application gives a member contained in any specified finite intersection. In particular every finite intersection is nonempty, so the family has the finite intersection property. Its upward closure

{A \subseteq X : \exists F \in F, F \subseteq A}

is the corresponding proper filter.

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