Star versions of the Menger property

Abstract

A space X is said to be star-Menger (resp., strongly star-Menger) if for each sequence {Un:n∈ω} of open covers of X, there are finite subfamilies Vn⊂Un (resp., finite subsets Fn⊂X) such that {St(⋃Vn,Un):n∈ω} (resp., {St(Fn,Un):n∈ω}) is a cover of X. These star versions of the Menger property were first introduced and studied in Kočinac [14,15]. In this paper, answering Song's 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.