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Ultrafilter


An ultrafilter on a nonempty set S is a nonempty collection F of subsets of S having the following properties:

1. ⌀ 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 set A^'=S-A in F.

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


See also

Cofinite Filter, Filter, Measurable Cardinal, Norm Ultrapower

This entry contributed by Viktor Bengtsson

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Cite this as:

Weisstein, Eric W., with contributions by Viktor Bengtsson. "Ultrafilter." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Ultrafilter.html

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