Hausdorff ultrafilters

We give the name Hausdorff to those ultrafilters that provide ultrapowers whose natural topology (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 can be Hausdorff. (u is the least size of an ultrafilter basis on ω.) We focus on countably incomplete ultrafilters, but our main results also hold for κ-complete ultrafilters.