Some observations on a clopen version of the Rothberger property

Abstract

In this paper, we proved that a clopen version $S_1(C_O, C_O)$ of the Rothberger property and Borel strong measure zeroness are independent. For a zero-dimensional metric space $(X, d)$, $X$ satisfies $S_1(C_O, C_O)$ if, and only if, $X$ has Borel strong measure zero with respect to each metric which has a same topology as d has. In a zero-dimensional space, the game $G_1(O,O)$ is equivalent to the game $G_1(C_O, C_O)$ and the point-open game is equivalent to the point-clopen game. Using reflections, we obtained that the game $G_1(C_O, C_O)$ and the point-clopen game are strategically and Markov dual. An example is given for a space on which the game $G_1(C_O, C_O)$ is undetermined.