SOME PROPERTIES OF ONE-POINT EXTENSIONS

Arkady Leiderman, Mikhail Tkachenko

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

A Tychonoff space Xp = X ∪ {p} is called a one-point extension of X if X is dense in Xp and the reminder Xp \X consists of the singleton {p}. We study the following problem: Characterize the spaces X such that every (some) one-point extension Xp of X has a given local topological property P at the point p. The list of properties P considered in the paper includes, among others: 1) {p} is a Gδ-set in Xp; 2) Xp admits a local countable base at p; 3) Xp has the Fréchet-Urysohn property at p; 4) Xp has countable tightness at p. One of our main results states that a Tychonoff space X is Lindelöf (not pseudocompact) iff the point p is of type Gδ in Xp, for every (for some, respectively) one-point extension Xp of X. We pose several open problems for various concrete properties P.

Original languageEnglish
Pages (from-to)195-208
Number of pages14
JournalTopology Proceedings
Volume59
Early online date8 May 2021
StatePublished - 1 Jan 2022

Keywords

  • Fréchet-Urysohn property
  • G-set
  • Lindelöf space
  • One-point extension
  • Stone-Čech compactification
  • character
  • zero-set

ASJC Scopus subject areas

  • Geometry and Topology

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