ω-cover, k-cover and related spaces on the Pixley-Roy PR[X]

Abstract

In this paper, we study the concepts of ω-cover, k-cover and certain spaces defined by them on the hyperspace PR[X] of finite subsets of a space X endowed with the Pixley-Roy topology. We prove that PR[X] is an ω-Rothberger (resp., ω-Menger, ω-Lindelöf) space if and only if X is countable. Moreover, we show that PR[X] is a k-Lindelöf and first-countable space if and only if X is a countable and first-countable space.