Van Douwen and many non Van Douwen families

Abstract

We prove that the spectrum of Van Douwen families is closed under singular limits. For any maximal eventually different family Raghavan defined in an associated ideal which measures how far the family is from being Van Douwen. Under CH we prove that every ideal containing Fin is realized as the associated ideal of some maximal eventually different family. Finally, we construct maximal eventually different families with Sacks-indestructible associated ideals to prove that in the iterated Sacks-model every $\aleph_1$-generated ideal containing Fin is realized.