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Topic: Principal Filter
Replies: 8   Last Post: Dec 7, 2013 8:31 PM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Principal Filter
Posted: Nov 24, 2013 3:59 AM
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For not empty A subset S,
let F_A be the principal filter for S generated by {A}.
For x in S,
let F_x = F_{x} be the principal ultrafilter for S generated by {{x}}.

Theorem. If F is a filter, then F = /\{ G ultrafilter | F subset G }
Proposition. If A not empty, then F_A = /\{ F_x | x in A }.

Question. If A not empty and F a filter with F_A subset F,
does F = /\{ F_x | x in A, F subset F_x }?

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