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.