Let S be a nonempty set, then an ultrafilter on S is a nonempty collection F of subsets of S having the following properties: 1.∅ not element F. 2. If A, B element F, then A intersection B element F. 3. If A element F and A⊆B⊆S, then B element F. 4. For any subset A of S, either A element F or its complement A' = S - A element F. An ultrafilter F on S is said to be free if it contains the cofinite filter F_S of S.
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