Monotone Normality and Nabla-Products

Abstract

Roitman's combinatorial principle $Δ$ is equivalent to monotone normality of the nabla product, $\nabla (ω+1)^ω$. If $\{ X_n : n\in ω\}$ is a family of metrizable spaces and $\nabla_n X_n$ is monotonically normal, then $\nabla_n X_n$ is hereditarily paracompact. Hence, if $Δ$ holds then the box product $\square (ω+1)^ω$ is paracompact. Large fragments of $Δ$ hold in $\mathsf{ZFC}$, yielding large subspaces of $\nabla (ω+1)^ω$ 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 $\nabla A(ω_1)^ω$ and $\nabla (ω_1+1)^ω$ are hereditarily normal (or hereditarily paracompact, or monotonically normal). In $\mathsf{ZFC}$ neither $\nabla A(ω_2)^ω$ nor $\nabla (ω_2+1)^ω$ is hereditarily normal.