A characterisation of compact, fragmentable linear orders

TL;DR

This paper characterizes compact, fragmentable linear orders, showing they are Radon-Nikodým compact and exploring implications in topology and renorming theory.

Contribution

It provides a new characterization of fragmentable compact linear orders and establishes their connection to Radon-Nikodým compact spaces.

Findings

Fragmentable compact linear orders are Radon-Nikodým compact.

Derived several corollaries in topology and renorming theory.

Established a characterization linking fragmentability and compactness in linear orders.

Abstract

We give a characterisation of fragmentable, compact linearly order spaces. In particular, we show that if $K$ is a compact, fragmentable, linearly ordered space then $K$ is a Radon-Nikod\'{y}m compact. In addition, we obtain some corollaries in topology and renorming theory.