Ultrafilters On ω
There are several special properties that an ultrafilter on ω may possess, which prove useful in various areas of set theory and topology.
- A non-principal ultrafilter U is a P-point (or weakly selective) iff for every partition of ω, such that, there exists such that .
- A non-principal ultrafilter U is Ramsey (or selective) iff for every partition of ω, such that, there exists such that
It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters. In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters. Therefore the existence of these types of ultrafilters is independent of ZFC.
P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non-principal ultrafilters. The name Ramsey comes from Ramsey's theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of there exists an element of the ultrafilter which has a homogeneous color.
An ultrafilter on ω is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal ultrafilters.
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