2020/06/26 by Barriga-Acosta, Hector A., Gartside, Paul M.
#FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2006.15163
Roitman's combinatorial principle Δ is equivalent to monotone normality of the nabla product, ∇ (ω+1)ω. If \ Xn : n∈ ω\ is a family of metrizable spaces and ∇n Xn is monotonically normal, then ∇n Xn is hereditarily paracompact. Hence, if Δ holds then the box product \square (ω+1)ω is paracompact. Large fragments of Δ hold in ZFC, yielding large subspaces of ∇ (ω+1)ω that are `really' monotonically normal. Countable nabla products of metrizable spaces which are respectively: arbitrary, of size ≤ \mathfrakc, or separable, are monotonically normal under respectively: \mathfrakb=\mathfrakd, \mathfrakd=\mathfrakc or the Model Hypothesis. It is consistent and independent that ∇ A(ω1)ω and ∇ (ω1+1)ω are hereditarily normal (or hereditarily paracompact, or monotonically normal). In ZFC neither ∇ A(ω2)ω nor ∇ (ω2+1)ω is hereditarily normal.