Diagonal conditions in ordered spaces

Harold R. Bennett, David J. Lutzer

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each T ⊂ X2 - Δ(X) with |T| = κ, there is an open neighborhood W of Δ(X) such that |T-W| = κ. If w1 ∈ D(X) then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems for such spaces, proving, for example, that a Lindelöf linearly ordered space with a small diagonal is metrizable. We give examples showing that our results are the sharpest possible, e.g., that there is a first countable, perfect, paracompact Čech-complete linearly ordered space with an H-diagonal that is not metrizable. Our example shows that a recent CH-result of Juhász and Szentmiklóssy on metrizability of compact Hausdorff spaces with small diagonals cannot be generalized beyond the class of locally compact spaces. We present examples showing the interplay of the above diagonal conditions with set theory in a natural extension of the Michael line construction.

Original languageEnglish
Pages (from-to)99-123
Number of pages25
JournalFundamenta Mathematicae
Volume153
Issue number2
StatePublished - 1997

Keywords

  • Cardinal invariant
  • Cech-complete space
  • Generalized ordered space
  • H-diagonal
  • Linearly ordered topological space
  • Metrizability
  • Michael line
  • P-space
  • Paracompact space
  • Small diagonal
  • Sorgenfrey line
  • σ-disjoint base

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