Madness and (weak) normality
Abstract
We consider weakenings of normality in -spaces and prove that the existence of a MAD family whose -space is almost-normal is independent of ZFC. We also construct a partly-normal not quasi-normal AD family, answering questions from Garc\'ia-Balan and Szeptycki. We finish by showing that the concepts of almost-normal and strongly 0-separated AD families are different, even under CH, answering a question from Oliveira-Rodrigues and Santos-Ronchim
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