Selectively strongly star-Menger spaces and related properties

Abstract

A space X is selectively strongly star-Menger (briefly, selSSM) if for each sequence (U_n : n ∈ N) of open covers of X and each sequence (D_n : n ∈ N) of dense subspaces of X, there exists a sequence (F_n : n ∈ N) of finite subsets F_n ⊂ D_n , n ∈ N, such that {St(F_n , U_n) : n ∈ N} is an open cover of X. This property is between absolute countable compactness [M. V. Matveev, Topol. Appl. 58, 81 (1994)] and selective absolute star-Lindelöfness [M. Bonanzinga et al., Topol. Appl. 221, 517 (2017)] and represents a "selective version" of the selection principle strongly star Menger [L. D. R. Kočinac, Publ. Math. Debrecen 55, 421 (1999)]. In this paper, we study some properties of selectively strongly star-Menger spaces, the relation with related properties and give some example distinguishing the properties considered.