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,...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Universidad de La Frontera
2023-08-01
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| Series: | Cubo |
| Subjects: | |
| Online Access: | https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3417/2296 |
| Summary: | 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. |
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| ISSN: | 0716-7776 0719-0646 |