When is the complement of the diagonal of a LOTS functionally countable?
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911833412075520 |
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| author | Gutiérrez-Domínguez, Luis Enrique Hernández-Gutiérrez, Rodrigo |
| author_facet | Gutiérrez-Domínguez, Luis Enrique Hernández-Gutiérrez, Rodrigo |
| contents | In a 2021 paper, Vladimir Tkachuk asked whether there is a non-separable LOTS $X$ such that $X^2\setminus\{\langle x,x\rangle\colon x\in X\}$ is functionally countable. In this paper we prove that such a space, if it exists, must be an Aronszajn line and admits a $\leq 2$-to-$1$ retraction to a subspace that is a Suslin line. After this, assuming the existence of a Suslin line, we prove that there is Suslin line that is functionally countable. Finally, we present an example of a functionally countable Suslin line $L$ such that $L^2\setminus\{\langle x,x\rangle\colon x\in L\}$ is not functionally countable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_14408 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | When is the complement of the diagonal of a LOTS functionally countable? Gutiérrez-Domínguez, Luis Enrique Hernández-Gutiérrez, Rodrigo General Topology 54F05, 06A05, 54A35, 54C30 In a 2021 paper, Vladimir Tkachuk asked whether there is a non-separable LOTS $X$ such that $X^2\setminus\{\langle x,x\rangle\colon x\in X\}$ is functionally countable. In this paper we prove that such a space, if it exists, must be an Aronszajn line and admits a $\leq 2$-to-$1$ retraction to a subspace that is a Suslin line. After this, assuming the existence of a Suslin line, we prove that there is Suslin line that is functionally countable. Finally, we present an example of a functionally countable Suslin line $L$ such that $L^2\setminus\{\langle x,x\rangle\colon x\in L\}$ is not functionally countable. |
| title | When is the complement of the diagonal of a LOTS functionally countable? |
| topic | General Topology 54F05, 06A05, 54A35, 54C30 |
| url | https://arxiv.org/abs/2211.14408 |