A classification of CO spaces which are continuous images of compact ordered spaces

TL;DR

This paper characterizes CO spaces that are continuous images of compact ordered spaces, showing they can be decomposed into simple summands, including certain ordinal-based spaces and a specific compactification.

Contribution

It provides an explicit classification of CO spaces that are continuous images of Dedekind complete totally ordered sets, expanding understanding of their structure.

Findings

Every member of the class K can be expressed as a finite disjoint sum of simple spaces.

Summands are either of the form mu + 1 + nu^* or the 1-point compactification of a discrete space with cardinality aleph_1.

The paper offers an explicit characterization of these CO spaces.

Abstract

A compact Hausdorff space X is called a CO space, if every closed subset of X is homeomorphic to an open subset of X. Every successor ordinal with its order topology is a CO space. We find an explicit characterization of the class K of CO spaces which are a continuous image of a Dedkind complete totally ordered set. (The topology of a totally ordered set is taken to be its order topology). We show that every member of K can be described as a finite disjoint sum of very simple spaces. Every summand has either form: (1) mu + 1 + nu^*, where mu and nu are cardinals, and nu^* is the reverse order of nu; or (2) the summand is the 1-point-compactification of a discrete space with cardinality aleph_1.