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Monotone normality and nabla products

Volume 254 / 2021

Hector A. Barriga-Acosta, Paul M. Gartside Fundamenta Mathematicae 254 (2021), 99-120 MSC: Primary 03E75, 54A35, 54B10, 54D15, 54D20; Secondary 54A25, 54B99, 54G20, 54G99. DOI: 10.4064/fm926-10-2020 Published online: 23 December 2020

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.

Authors

  • Hector A. Barriga-AcostaPosgrado Conjunto en Ciencias Matemáticas
    UMSNH-UNAM
    Morelia, Mexico
    e-mail
  • Paul M. GartsideDepartment of Mathematics
    University of Pittsburgh
    Pittsburgh, PA, U.S.A.
    e-mail

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