Some observations on a clopen version of the Rothberger property




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.


Strong measure zero , selection principles , point-clopen game , zero-dimensional space

Mathematics Subject Classification:

54D20 , 54A20
  • Pages: 161–172
  • Date Published: 2023-07-19
  • Vol. 25 No. 2 (2023)

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  • Ministry of Science and Higher Education of the Russian Federation


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How to Cite

M. Bhardwaj and A. V. Osipov, “Some observations on a clopen version of the Rothberger property”, CUBO, pp. 161–172, Jul. 2023.