Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hrušák
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918303821201408 |
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| author | Medini, Andrea |
| author_facet | Medini, Andrea |
| contents | Given a cardinal $κ$ and filters $\mathcal{F}_α$ on $ω$ for $α\inκ$, we will show that if $\prod_{α\inκ}\mathcal{F}_α$ is countable dense homogeneous then $κ<\mathfrak{p}$ and each $\mathcal{F}_α$ is a non-meager $\mathsf{P}$-filter. This partially answers a question of Michael Hrušák. Along the way, we will show that the product of fewer than $\mathfrak{p}$ non-meager $\mathsf{P}$-filters has the Miller property. We will also describe explicitly the connection between Miller-measurability and the Miller property. As a corollary, we will see that the intersection of fewer than $\mathsf{add}(m^0)$ non-meager $\mathsf{P}$-filters is a non-meager $\mathsf{P}$-filter, where $m^0$ denotes the ideal of Miller-null sets. We will conclude by investigating the preservation of the Miller property under intersections and products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17780 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hrušák Medini, Andrea Logic General Topology 03E05, 03E17, 54B10 Given a cardinal $κ$ and filters $\mathcal{F}_α$ on $ω$ for $α\inκ$, we will show that if $\prod_{α\inκ}\mathcal{F}_α$ is countable dense homogeneous then $κ<\mathfrak{p}$ and each $\mathcal{F}_α$ is a non-meager $\mathsf{P}$-filter. This partially answers a question of Michael Hrušák. Along the way, we will show that the product of fewer than $\mathfrak{p}$ non-meager $\mathsf{P}$-filters has the Miller property. We will also describe explicitly the connection between Miller-measurability and the Miller property. As a corollary, we will see that the intersection of fewer than $\mathsf{add}(m^0)$ non-meager $\mathsf{P}$-filters is a non-meager $\mathsf{P}$-filter, where $m^0$ denotes the ideal of Miller-null sets. We will conclude by investigating the preservation of the Miller property under intersections and products. |
| title | Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hrušák |
| topic | Logic General Topology 03E05, 03E17, 54B10 |
| url | https://arxiv.org/abs/2601.17780 |