## Abstract

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_{n}has 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 S_{1}(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 language | English |
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Article number | 108257 |

Journal | Topology and its Applications |

Volume | 321 |

DOIs | |

State | Published - 1 Nov 2022 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2022 Elsevier B.V.

## Keywords

- Conjecture
- Problem
- Selection property
- γ-Cover