The finer selection properties

For a function f from ω to ω, a topological space X satisfies ⋃_f(Γ,Γ) if for each sequence (U_n:n∈ω,U_nhas no finite subcover) of elements of Γ, select for each n a finite subset F_n⊆U_n such that |F_n|≤f(n) for all n and {⋃F_n: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.