Semi-stratifiable spaces with monotonically normal compactifications

Harold Bennett, David Lutzer

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

In this paper we use Mary Ellen Rudin's solution of Nikiel's problem to investigate metrizability of certain subsets of compact monotonically normal spaces. We prove that if H is a semi-stratifiable space that can be covered by a σ-locally-finite collection of closed metrizable subspaces and if H embeds in a monotonically normal compact space, then H is metrizable. It follows that if H is a semi-stratifiable space with a monotonically normal compactification, then H is metrizable if it satisfies any one of the following: H has a σ-locally finite cover by compact subsets; H is a σ-discrete space; H is a scattered; H is σ-compact. In addition, a countable space X has a monotonically normal compactification if and only if X is metrizable. We also prove that any semi-stratifiable space with a monotonically normal compactification is first-countable and is the union of a family of dense metrizable subspaces. Having a monotonically normal compactification is a crucial hypothesis in these results because R.W. Heath has given an example of a countable non-metrizable stratifiable (and hence monotonically normal) group. We ask whether a first-countable semi-stratifiable space must be metrizable if it has a monotonically normal compactification. This is equivalent to "If X is a first-countable stratifiable space with a monotonically normal compactification, must H be metrizable?".

Original languageEnglish
Pages (from-to)21-27
Number of pages7
JournalTopology and its Applications
Volume197
DOIs
StatePublished - Jan 1 2016

Keywords

  • Countable subspace
  • Dense metrizable subset
  • Metrizable space
  • Monotonically normal compactification
  • Monotonically normal space
  • Rudin's solution of Nikiel's problem
  • Scattered
  • Semi-stratifiable
  • Stratifiable
  • σ-Discrete space

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