Some observations on a clopen version of the Rothberger property

Abstract

In this paper, we prove that a clopen version $S_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ of the Rothberger property and Borel strong measure zeroness are independent. For a zero-dimensional metric space $(X,d)$, $X$ satisfies $S_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ 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 $G_1(\mathcal{O}, \mathcal{O})$ is equivalent to the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ and the point-open game is equivalent to the point-clopen game. Using reflections, we obtain that the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ and the point-clopen game are strategically and Markov dual. An example is given for a space on which the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ is undetermined.