DIAMOND on LADDER SYSTEMS and COUNTABLY METACOMPACT TOPOLOGICAL SPACES

Rodrigo Carvalho, Tanmay Inamdar, A. S.S.A.F. Rinot

Research output: Contribution to journalArticlepeer-review

Abstract

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval [0; 1] is again normal i δ X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied δ-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system L over the first uncountable cardinal for which the corresponding space XL is not a δ-space, and asked whether there is a ZFC example of a ladder system L over some cardinal δ for which XL is not countably metacompact, in particular, not a δ-space. We prove that an δ rmative answer holds for the cardinal ]ω+ = cf(]ω+1). Assuming i! =, we get an example at a much lower cardinal, namely κ = 222ℵ0, and our ladder system L is moreover !-bounded.

Original languageEnglish
JournalJournal of Symbolic Logic
DOIs
StateAccepted/In press - 2024

Bibliographical note

Publisher Copyright:
© 2024 Cambridge University Press. All rights reserved.

Keywords

  • Countably metacompact
  • Ladder system
  • Middle diamond
  • ZFC combinatorics
  • δ-space
  • Ψ-space

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