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. emptyset not in F.

2. If A,B in F, then A intersection B in F.

3. If A in F and A subset= B subset= S, then B in F.

4. For any subset A of S, either A in F or its complement A^'=S-A in F.

An ultrafilter F on S is said to be free if it contains the cofinite filter F_S of S.

See also

Cofinite Filter, Filter

This entry contributed by Viktor Bengtsson

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Bengtsson, Viktor. "Ultrafilter." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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