On spaces with star kernel Menger

Given a topological property $\mathcal{P}$, a space $X$ is called star-$\mathcal{P}$ if for any open cover $\mathcal{U}$ of the space $X$, there exists a set $Y\subseteq X$ with property $\mathcal{P}$ such that $St(Y,\mathcal{U})=X$; the set $Y$ is called a star kernel of the cover $\mathcal{U}$. In this paper, we introduce and study spaces with star kernel Menger, that is, star Menger spaces. Some examples are given to show the relationship with some other related properties studied previously, and the behaviour of the star Menger property with respect to subspaces, products, continuous images and preimages are investigated. Additionally, some comments on the star selection theory are given. Particularly, some questions posed by Song within this theory are addressed. Finally, several new properties are introduced as well as some general questions on them are posed.