$\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916890920615936 |
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| author | Brian, Will Dow, Alan Hart, Klaas Pieter |
| author_facet | Brian, Will Dow, Alan Hart, Klaas Pieter |
| contents | Let $\mathbb{M} = \mathbb N \times [0,1]$. The natural projection $π: \mathbb{M} \rightarrow \mathbb N$, which sends $(n,x)$ to $n$, induces a projection mapping $π^*: \mathbb{M}^* \rightarrow \mathbb N^*$, where $\mathbb{M}^*$ and $\mathbb N^*$ denote the Čech-Stone remainders of $\mathbb{M}$ and $\mathbb N$, respectively.
We show that $\mathsf{CH}$ implies every autohomeomorphism of $\mathbb N^*$ lifts through the natural projection to an autohomeomorphism of $\mathbb{M}^*$. That is, for every homeomorphism $h: \mathbb N^* \rightarrow \mathbb N^*$ there is a homeomorphism $H: \mathbb{M}^* \rightarrow \mathbb{M}^*$ such that $π^* \circ H = h \circ π^*$. This complements a recent result of the second author, who showed that this lifting property is not a consequence of $\mathsf{ZFC}$.
Combining this lifting theorem with a recent result of the first author, we also prove that $\mathsf{CH}$ implies there is an order-reversing autohomeomorphism of~$\mathbb H^*$, the Čech-Stone remainder of the half line $\mathbb H = [0,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04425 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$ Brian, Will Dow, Alan Hart, Klaas Pieter General Topology Logic primary: 54D40, 03E35, secondary: 06D50, 3C20 Let $\mathbb{M} = \mathbb N \times [0,1]$. The natural projection $π: \mathbb{M} \rightarrow \mathbb N$, which sends $(n,x)$ to $n$, induces a projection mapping $π^*: \mathbb{M}^* \rightarrow \mathbb N^*$, where $\mathbb{M}^*$ and $\mathbb N^*$ denote the Čech-Stone remainders of $\mathbb{M}$ and $\mathbb N$, respectively. We show that $\mathsf{CH}$ implies every autohomeomorphism of $\mathbb N^*$ lifts through the natural projection to an autohomeomorphism of $\mathbb{M}^*$. That is, for every homeomorphism $h: \mathbb N^* \rightarrow \mathbb N^*$ there is a homeomorphism $H: \mathbb{M}^* \rightarrow \mathbb{M}^*$ such that $π^* \circ H = h \circ π^*$. This complements a recent result of the second author, who showed that this lifting property is not a consequence of $\mathsf{ZFC}$. Combining this lifting theorem with a recent result of the first author, we also prove that $\mathsf{CH}$ implies there is an order-reversing autohomeomorphism of~$\mathbb H^*$, the Čech-Stone remainder of the half line $\mathbb H = [0,\infty)$. |
| title | $\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$ |
| topic | General Topology Logic primary: 54D40, 03E35, secondary: 06D50, 3C20 |
| url | https://arxiv.org/abs/2505.04425 |