There May Be No Nowhere Dense Ultrafilter

Abstract

We show the consistency of ZFC +''there is no NWD-ultrafilter on omega'', which means: for every non principle ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for some nowhere dense set A of reals, the set {n: f(n) in A} is not in D. This answers a question of van Douwen, which was put in more general context by Baumgartner