Monotone normality and nabla products
Volume 254 / 2021
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.
