Images of the countable ordinals

Harold Bennett, Sheldon Davis, David Lutzer

Research output: Contribution to journalArticle

Abstract

We study spaces that are continuous images of the usual space [0,ω1) of countable ordinals. We begin by showing that if Y is such a space and is T3 then Y has a monotonically normal compactification, and is monotonically normal, locally compact and scattered. Examples show that regularity is needed in these results. We investigate when a regular continuous image of the countable ordinals must be compact, paracompact, and metrizable. For example we show that metrizability of such a Y is equivalent to each of the following: Y has a Gδ-diagonal, Y is perfect, Y has a point-countable base, Y has a small diagonal in the sense of Hušek, and Y has a σ-minimal base. Along the way we obtain an absolute version of the Juhasz–Szentmiklossy theorem for small spaces, proving that if Y is any compact Hausdorff space having |Y|≤ℵ1 and having a small diagonal, then Y is metrizable, and we deduce a recent result of Gruenhage from work of Mrowka, Rajagopalan, and Soundararajan.

Original languageEnglish
Pages (from-to)610-623
Number of pages14
JournalTopology and its Applications
Volume221
DOIs
StatePublished - Apr 15 2017

Keywords

  • Compact
  • Compact Hausdorff space with cardinality ℵ
  • Continuous images of the countable ordinals
  • Countable ordinals
  • Juhasz–Szentmiklossy theorem
  • Locally compact
  • Metrizable
  • Monotonically normal
  • Monotonically normal compactification
  • Paracompact
  • Scattered
  • Small diagonals
  • σ-Minimal base

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