The finer selection properties

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.