Hausdorff ultrafilters

Abstract

We give the name Hausdorff to those ultrafilters that provide ultrapowers whose natural topology ( S S -topology) is Hausdorff, e.g. selective ultrafilters are Hausdorff. Here we give necessary and sufficient conditions for product ultrafilters to be Hausdorff. Moreover we show that no regular ultrafilter over the “small” uncountable cardinal u \mathfrak {u} can be Hausdorff. ( u \mathfrak {u} is the least size of an ultrafilter basis on ω \omega .) We focus on countably incomplete ultrafilters, but our main results also hold for κ \kappa -complete ultrafilters.