Partitioning the real line into Borel sets
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910453227061248 |
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| author | Brian, Will |
| author_facet | Brian, Will |
| contents | For which infinite cardinals $κ$ is there a partition of the real line $\mathbb R$ into precisely $κ$ Borel sets?
Hausdorff famously proved that there is a partition of $\mathbb R$ into $\aleph_1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of $\mathbb R$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq ω$ with $0,1 \in A$, there is a forcing extension in which $A = \{ n :\, \text{there is a partition of }\mathbb R\text{ into }\aleph_n\text{ Borel sets}\}$.
We also look at the corresponding question for partitions of $\mathbb R$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $κ$ such that there is a partition of $\mathbb R$ into precisely $κ$ closed sets can be fairly arbitrary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_00535 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Partitioning the real line into Borel sets Brian, Will Logic For which infinite cardinals $κ$ is there a partition of the real line $\mathbb R$ into precisely $κ$ Borel sets? Hausdorff famously proved that there is a partition of $\mathbb R$ into $\aleph_1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of $\mathbb R$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq ω$ with $0,1 \in A$, there is a forcing extension in which $A = \{ n :\, \text{there is a partition of }\mathbb R\text{ into }\aleph_n\text{ Borel sets}\}$. We also look at the corresponding question for partitions of $\mathbb R$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $κ$ such that there is a partition of $\mathbb R$ into precisely $κ$ closed sets can be fairly arbitrary. |
| title | Partitioning the real line into Borel sets |
| topic | Logic |
| url | https://arxiv.org/abs/2112.00535 |