Almost-normality of Isbell-Mr\'owka spaces

TL;DR

This paper investigates the properties of almost-normality in Isbell-Mrówka spaces, providing new examples, exploring related concepts like semi-normality, and establishing the existence of certain strongly separated almost disjoint families under set-theoretic assumptions.

Contribution

It introduces new examples of almost-normal but not normal spaces, defines strongly separated almost disjoint families, and answers an open question about pseudocompact spaces.

Findings

Constructed an almost-normal not normal almost disjoint family using forcing.

Defined and analyzed strongly $(eth_0, <eth_1)$-separated almost disjoint families.

Provided an example of a Tychonoff almost-normal not normal pseudocompact space that is not countably compact.

Abstract

We explore almost-normality in Isbell-Mr\'owka 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-Mr\'owka 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 which is not countably compact, answering a question from P. Szeptycki and S. Garcia-Balan.