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

TL;DR

This paper investigates the conditions under which the complement of the diagonal in a non-separable linearly ordered topological space (LOTS) is functionally countable, revealing connections to Suslin lines and providing a specific counterexample.

Contribution

It characterizes the structure of non-separable LOTS with functionally countable off-diagonal complements and constructs a counterexample assuming the existence of a Suslin line.

Findings

Such spaces must be Aronszajn lines with a specific retraction property.

Under the assumption of a Suslin line, a Suslin line can be functionally countable.

Counterexample of a functionally countable Suslin line with a non-functionally countable off-diagonal complement.

Abstract

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.