When is the complement of the diagonal of a LOTS functionally countable?

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 t...

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Published in:arXiv.org 2024-04
Main Authors: Gutiérrez-Domínguez, Luis Enrique, Hernández-Gutiérrez, Rodrigo
Format: Article
Language:English
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2211.14408