On the first Banach problem, concerning condensations of absolute $κ$-Borel sets onto compacta
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913238834216960 |
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| author | Osipov, Alexander V. |
| author_facet | Osipov, Alexander V. |
| contents | It is consistent that the continuum be arbitrary large and no absolute $κ$-Borel set $X$ of density $κ$, $\aleph_1<κ<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large and any absolute $κ$-Borel set $X$ of density $κ$, $κ\leq\mathfrak{c}$, containing a closed subspace of the Baire space of weight $κ$, condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any $A\subseteq \mathbb{N}$ with $1\in A$, there is a forcing extension in which every absolute $\aleph_n$-Borel set, containing a closed subspace of the Baire space of weight $\aleph_n$, condenses onto a compactum if, and only if, $n\in A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_05942 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the first Banach problem, concerning condensations of absolute $κ$-Borel sets onto compacta Osipov, Alexander V. General Topology It is consistent that the continuum be arbitrary large and no absolute $κ$-Borel set $X$ of density $κ$, $\aleph_1<κ<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large and any absolute $κ$-Borel set $X$ of density $κ$, $κ\leq\mathfrak{c}$, containing a closed subspace of the Baire space of weight $κ$, condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any $A\subseteq \mathbb{N}$ with $1\in A$, there is a forcing extension in which every absolute $\aleph_n$-Borel set, containing a closed subspace of the Baire space of weight $\aleph_n$, condenses onto a compactum if, and only if, $n\in A$. |
| title | On the first Banach problem, concerning condensations of absolute $κ$-Borel sets onto compacta |
| topic | General Topology |
| url | https://arxiv.org/abs/2209.05942 |