AVOIDING FAMILIES AND TUKEY FUNCTIONS ON THE NOWHERE-DENSE IDEAL

We investigate Tukey functions from the ideal of all closed nowhere-dense subsets of 2(N). In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of N. We also prove non-existence of various special types of Tukey reductions from the nowhere-dense ideal to analytic P-ideals. In connection with these results, we study families F of clopen subsets of 2(N) with the property that for each nowhere-dense subset of 2(N) there is a set in F not intersecting it. We call such families avoiding.