On properties defined via certain Menger-type kernels

Given a topological property P P , a space X is called star- P P if for any open cover U U of the space X , there exists a set Y ⊆ X satisfying the property P P such that S t ( Y , U ) = X St(Y,U)=X ; the set Y is called a star kernel of the cover U U . In (J. Casas-de la Rosa and Á. Tamariz-Mascarúa, “On spaces with star kernel Menger,” Acta Math. Hung. , vol. 170, pp. 379–404, 2023), the authors introduced and studied spaces with a star kernel Menger. Motivated by this idea, in this paper we introduced several properties obtained by requiring certain kernel with Menger property. Namely, the dually Menger spaces and the cellular Menger spaces as well as some of the almost and weak versions of them. Some examples are given to show the relationship among these properties. Additionally, we give some results on the Pixley–Roy hyperspace that involve some of these properties. Finally, we make some comments on a Song’s example given in (Y. K. Song, “A pseudocompact Tychonoff space that is not star Lindelöf,” Bull. Aust. Math. Soc. , vol. 84, pp. 452–454, 2011).