Separable linear orders and universality

TL;DR

The paper clarifies a misconception in the literature by showing that not all separable linear orders embed into the real line under the usual definition of separability.

Contribution

It corrects the false claim that all separable linear orders embed into the real line, providing a precise understanding of their relationship.

Findings

Not all separable linear orders embed into the real line.

The common assumption about separable linear orders embedding into the real line is false.

Clarifies the correct conditions for embedding of separable linear orders.

Abstract

In various places in the literature it is stated that every separable linear order embeds into the real line. This is, however, not the case, at least not with respect to the usual definition of separability. We correct this misconception.