Cleavability and scattered sets of non-trivial fibers

TL;DR

This paper investigates conditions under which a compact space cleavable over a linearly ordered topological space (LOTS) must itself be a LOTS, focusing on the nature of non-injective points of continuous functions.

Contribution

It provides a partial answer to a problem by Arhangel'skii, showing that certain cleavability conditions imply the space is a LOTS.

Findings

If a compactum is cleavable over a separable LOTS and has a continuous function with scattered non-injective points, then it is a LOTS.

The paper characterizes when a space cleavable over a LOTS must itself be a LOTS.

It advances understanding of the structure of cleavable spaces in topology.

Abstract

In this paper, we provide a partial answer to a problem posed by A. V.Arhangel'skii; we show that if X is a compactum cleavable over a separable linearly ordered topological space (LOTS) Y such that for some continuous function f from X to Y, the set of points on which f is not injective is scattered, then X is a LOTS.