Star versions of the Menger property

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