Some observations on a clopen version of the Rothberger property

Abstract

In this paper, we prove that a clopen version S1(CO, CO) of the Rothberger property and Borel strong measure ze-roness are independent. For a zero-dimensional metric space (X, d), X satisfies S1(CO, CO) if, and only if, X has Borel strong measure zero with respect to each metric which has the same topology as d has. In a zero-dimensional space, the game G1(O, O) is equivalent to the game G1(CO, CO) and the point-open game is equivalent to the point-clopen game. Us-ing reflections, we obtain that the game G1(CO, CO) and the point-clopen game are strategically and Markov dual. An example is given for a space on which the game G1(CO, CO) is undetermined. © 2023 Manoj Bhardwaj et al.