A note on set-star-K-Menger spaces

Abstract

Abstract A space X is said to have the set-star-K-Menger property if for each nonempty subset A of X and for each sequence (𝓤 n : n ∈ ℕ) of collections of open sets in X such that for each n ∈ ℕ, A ⊆ ⋃ 𝓤 n , there is a sequence (Kn : n ∈ ℕ) of compact subsets of X such that A ⊆ $\begin{array}{} \bigcup\limits_{n \in \mathbb{N}} \end{array} $ St(Kn , 𝓤 n ). In this paper, we prove that: There exists a T 1 set-star-Menger space which is not set-star-K-Menger and there exists a Tychonoff set-star-K-Menger space that is not set-star-Menger. Assuming 𝔡 = 𝔠, there exists a Tychonoff set-star-K-Menger space having a regular-closed Gδ -subspace which is not set-star-K-Menger. If the Alexandroff duplicate of a space X is set-star-K-Menger, then X is set-star-K-Menger. The product of set-star-K-Menger space and a compact space is rectangular set-star-K-Menger space. The above-mentioned results answer to Problem 2.5 and Problem 3.6, and give a partial answer to Problem 3.11 in [SINGH, S.: On set-star-K-Menger spaces, Publ. Math. Debrecen 100 (2022), 87–100]. Further, we continue to study the topological properties of set-star-K-Menger spaces.