The finer selection properties

Meiyan Liu, Jialiang He, Shuguo Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

For a function f from ω to ω, a topological space X satisfies ⋃f(Γ,Γ) if for each sequence (Un:n∈ω,Unhas no finite subcover) of elements of Γ, select for each n a finite subset Fn⊆Un such that |Fn|≤f(n) for all n and {⋃Fn:n∈ω} is an element of Γ, where Γ denotes the family of all open γ-covers of X. In this paper, we prove the following results. (1) Assume the Continuum Hypothesis. There is a set of real numbers that satisfies ⋃fin(Γ,Γ) and S1(Γ,O) but not ⋃id(Γ,Γ), where id is the identity function from ω to ω. (2) Assume b=c. There is a set of real numbers that satisfies ⋃id(Γ,Γ) but not ⋃k(Γ,Γ) for all natural numbers k≥1. (3) Assume the Continuum Hypothesis. For each natural number k≥2, there is a set of real numbers that satisfies ⋃k+1(Γ,Γ) but not ⋃k(Γ,Γ). These results answer an open problem proposed by Zdomskyy and a conjecture proposed by Tsaban.

Original languageEnglish
Article number108257
JournalTopology and its Applications
Volume321
DOIs
StatePublished - 1 Nov 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2022 Elsevier B.V.

Keywords

  • Conjecture
  • Problem
  • Selection property
  • γ-Cover

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