Linearly-ordered Radon-Nidkodym compact spaces

We prove that every fragmentable linearly ordered compact space is almost totally disconnected. This combined with a result of Arvanitakis yields that every linearly ordered quasi Radon-Nikodym compact space is Radon-Nikodym, providing a new partial answer to the problem of continuous images of Radon-Nikodym compacta.

💡 Research Summary

The paper addresses the long‑standing question of whether the class of Radon‑Nikodym compact spaces is closed under continuous images. It focuses on the subclass of linearly ordered compact spaces and establishes a new partial answer. The authors first recall the standard characterizations: a compact space K is Radon‑Nikodym (RN) if there exists a lower‑semicontinuous metric d that fragments K; K is quasi‑Radon‑Nikodym (QRN) if a lower‑semicontinuous quasi‑metric (a symmetric, zero‑only‑on‑the‑diagonal map that may fail the triangle inequality) fragments K; and K is fragmentable if some quasi‑metric (not required to be lower‑semicontinuous) fragments K.

A central notion introduced is that of “almost totally disconnected” compact spaces. Such a space is homeomorphic to a subset of Σ₁⁰