Cleavability over ordinals

TL;DR

This paper proves that infinite compact spaces cleavable over an ordinal are homeomorphic to that ordinal, resolving key questions about their structure and embedding properties.

Contribution

It establishes that such spaces are necessarily homeomorphic to ordinals and are linearly ordered, answering fundamental questions in cleavability theory.

Findings

Infinite compacta cleavable over an ordinal are homeomorphic to the ordinal.

Such spaces are necessarily linearly ordered (LOTS).

Open question remains on embeddability into the ordinal.

Abstract

In this paper we show that if X is an infinite compactum cleavable over an ordinal, then X must be homeomorphic to an ordinal. X must also therefore be a LOTS. This answers two fundamental questions in the area of cleavability. We also leave it as an open question whether cleavability of an infinite compactum X over an ordinal \lambda implies X is embeddable into \lambda.