Monotone Normality and Nabla-Products

TL;DR

This paper explores the relationship between a combinatorial principle and the monotone normality of nabla products, establishing conditions under which these products are hereditarily paracompact and normal, with results depending on set-theoretic assumptions.

Contribution

It links a combinatorial principle to the monotone normality of nabla products and provides conditions for hereditarily normal and paracompact properties in various set-theoretic contexts.

Findings

Roitman's principle $ abla$ is equivalent to monotone normality of $ abla ( ext{omega}+1)^ ext{omega}$.

Under certain set-theoretic assumptions, nabla products of metrizable spaces are monotonically normal.

Certain nabla products are hereditarily normal or paracompact depending on set-theoretic hypotheses.

Abstract

Roitman's combinatorial principle $\Delta$ is equivalent to monotone normality of the nabla product, $\nabla (\omega +1)^\omega$. If $\{ X_n : n\in \omega\}$ is a family of metrizable spaces and $\nabla_n X_n$ is monotonically normal, then $\nabla_n X_n$ is hereditarily paracompact. Hence, if $\Delta$ holds then the box product $\square (\omega +1)^\omega$ is paracompact. Large fragments of $\Delta$ hold in $\mathsf{ZFC}$, yielding large subspaces of $\nabla (\omega+1)^\omega$ that are `really' monotonically normal. Countable nabla products of metrizable spaces which are respectively: arbitrary, of size $\le \mathfrak{c}$, or separable, are monotonically normal under respectively: $\mathfrak{b}=\mathfrak{d}$, $\mathfrak{d}=\mathfrak{c}$ or the Model Hypothesis. It is consistent and independent that $\nabla A(\omega_1)^\omega$ and $\nabla (\omega_1+1)^\omega$ are hereditarily normal (or hereditarily paracompact, or monotonically normal). In $\mathsf{ZFC}$ neither $\nabla A(\omega_2)^\omega$ nor $\nabla (\omega_2+1)^\omega$ is hereditarily normal.