On set-star-K-Menger spaces

A topological space $X$ is said to have the set-star-K-Menger property if for each nonempty subset $A$ of $X$ and for each sequence $(\mathcal{U}_n:n\in\mathbb{N})$ of open families in $X$ such that $\overline{A}\subseteq\bigcup\mathcal{U}_n$ for all $n\in\mathbb{N}$, there is a sequence $(K_n:n\in\mathbb{N})$ of compact subsets of $X$ such that $A\subseteq\bigcup_{n\in\mathbb{N}}\operatorname{St}(K_n,\mathcal{U}_n)$. This property is motivated by the sLindelöf cardinal function in Arhangel'skii [1] and set-star covering properties introduced by Kočinac, Konca and Singh [10]. We investigate the relationships between the set-star-K-Menger and other related properties, and study the topological properties of the set-star-K-Menger property.