Monotone Normality and Nabla-Products

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.