Poorly separated infinite normal products

Madness and (weak) normality

We consider weakenings of normality in $Ψ$-spaces and prove that the existence of a MAD family whose $Ψ$-space is almost-normal is independent of \textsf{ZFC}. We also construct a partly-normal not quasi-normal AD family, answering questions from García-Balan and Szeptycki. We finish by showing that the concepts of almost-normal and strongly $\aleph_0$-separated AD families are different, even under \textsf{CH}, answering a question from Oliveira-Rodrigues and Santos-Ronchim

Almost-normality of Isbell-Mrówka spaces

We explore almost-normality in Isbell-Mrówka spaces and some related concepts. We use forcing to provide an example of an almost-normal not normal almost disjoint family, explore the concept of semi-normality in Isbell-Mrówka spaces, define the concept of strongly $(\aleph_0, <\mathfrak c)$-separated almost disjoint families and prove the generic existence of completely separable strongly $(\aleph_0, <\mathfrak c)$-almost disjoint families assuming $\mathfrak s=\mathfrak c$ and $\mathfrak b=\mathfrak c$. We also provide an example of a Tychonoff almost-normal not normal pseudocompact space whi…

通过Urysohn引理的提升性质来定义正常空间

Normal Spaces via Urysohn's Lemma as a Lifting Property

本文通过Urysohn引理的提升性质重新定义正常空间，并将继承正常空间的定义转化为开子空间的正常性条件。

Special sets of reals and weak forms of normality on Isbell-Mrówka spaces

We recall some classical results relating normality and some natural weakenings of normality in $Ψ$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $λ$-sets and $σ$-sets. We introduce a new class of special sets of reals which corresponds the corresponding almost disjoint family of branches being $\aleph_0$-separated. This new class fits between $λ$-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $\mathcal A$ being potentially almost-normal (pseudonormal), in the sense that $\mathcal A$ is almo…