Weak normality properties in $Ψ$-spaces

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

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