Weak normality properties in $\Psi$-spaces

Abstract

Almost disjoint families of true cardinality $\mathfrak{c}$ are used to produce an example of a mildly-normal not partly-normal $\Psi$-space and a quasi-normal not almost-normal $\Psi$-space. This is related with a problem posed by Lufti Kalantan where he asks whether there exists a mad family so that the related Mr\'owka-Isbell space is partly-normal. In addition, a consistent example of a Luzin mad family such that its associated $\Psi$-space is quasi-normal is provided.