Van Douwen and many non Van Douwen families

TL;DR

This paper explores the properties of Van Douwen families and their associated ideals, demonstrating closure under limits, realizability of ideals, and indestructibility in certain models, advancing understanding of maximal eventually different families.

Contribution

It establishes the closure of Van Douwen families under singular limits, characterizes ideals associated with maximal eventually different families, and constructs indestructible families in the Sacks-model.

Findings

Spectrum of Van Douwen families is closed under singular limits

Every ideal containing Fin can be realized as an associated ideal under CH

Constructs Sacks-indestructible maximal eventually different 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.