Star versions of the Menger property

A space X is said to be star-Menger (resp., strongly star-Menger) if for each sequence { U n : n ∈ ω } of open covers of X, there are finite subfamilies V n ⊂ U n (resp., finite subsets F n ⊂ X ) such that { St ( ⋃ V n , U n ) : n ∈ ω } (resp., { St ( F n , U n ) : n ∈ ω } ) is a cover of X. These star versions of the Menger property were first introduced and studied in Kočinac [14,15]. In this paper, answering Songʹs question, we show that the extent of a regular strongly star-Menger space cannot exceed the continuum c . Star-Menger Pixley–Roy hyperspaces PR ( X ) are also investigated. We show that if a space X is regular and PR ( X ) is star-Menger, then the cardinality of X is less than c and every finite power of X is Menger.