Monotone normality and nabla products
Volume 254 / 2021
Abstract
Roitman’s combinatorial principle 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.