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

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.

💡 Research Summary

The paper investigates a very specialized class of compact Hausdorff spaces known as CO spaces—spaces in which every closed subset is homeomorphic to an open subset of the same space. While it is classical that every successor ordinal equipped with its order topology is a CO space, a systematic description of CO spaces that arise as continuous images of Dedekind‑complete totally ordered sets (i.e., linearly ordered sets in which every non‑empty bounded subset has a supremum and an infimum) has been missing.

The authors begin by recalling that a Dedekind‑complete linear order D, endowed with the order topology, is itself a compact Hausdorff space. A continuous surjection f : D → X therefore forces X to inherit much of the “interval” structure of D, yet X may acquire additional limit points or “glue points” that are not present in D. The CO condition imposes a strong self‑similarity: for any closed set A⊆X there must exist an open set U⊆X such that A≅U (homeomorphic). This forces a separation property that mirrors the way intervals in D are separated by their endpoints.

The central technical achievement is a decomposition theorem. The authors first show that any CO space X that is a continuous image of a Dedekind‑complete order can be written as a finite disjoint sum of its connected components, each of which is itself a CO space. Because each component must still be a continuous image of a sub‑order of D, it inherits a linear order‑like structure. By analysing the possible configurations of closed sets and their corresponding open counterparts, the authors prove that each component can only be of one of two canonical forms:

1. μ + 1 + ν* – Here μ and ν are cardinals (possibly infinite). The notation μ + 1 denotes an initial segment of order type μ followed by a single isolated point, while ν* denotes the reverse order of a segment of type ν. The central “+1” point acts as a limit point that ties the two monotone sides together. This form captures all spaces that look like a two‑sided linear order with a distinguished “pivot” point.

2. One‑point compactification of a discrete space of size ℵ₁ – Take a discrete set of cardinality ℵ₁ (the first uncountable cardinal) and add a single extra point whose neighborhoods are co‑finite in the discrete part. The resulting space is compact, Hausdorff, and satisfies the CO property even though the underlying discrete part alone does not.

The proof that no other configurations can occur relies on a careful interplay between cardinal arithmetic, the Dedekind completeness of the source order, and the rigidity imposed by the CO condition. In particular, any attempt to introduce more than finitely many “glue points” would break compactness or the requirement that every closed set be homeomorphic to an open set. Likewise, any component that is not of the two forms above would either fail to be a continuous image of a Dedekind‑complete order or would violate the CO property by producing a closed set without a matching open counterpart.

Having identified the two building blocks, the authors then show that any admissible CO space X can be expressed as a finite disjoint union (topological sum) of spaces each of which is either of type (1) or type (2). The finiteness of the sum is essential: an infinite disjoint sum would lose compactness, while a non‑disjoint arrangement would introduce new limit points that cannot be accounted for by the two canonical forms.

Consequently, the class K of CO spaces that are continuous images of compact ordered spaces is completely characterized:

• Every member of K is a finite topological sum of spaces of the form μ + 1 + ν* (μ, ν cardinals) or the one‑point compactification of a discrete space of cardinality ℵ₁.
• Conversely, any finite sum of such spaces is a CO space and can be realized as a continuous image of a suitable Dedekind‑complete linear order.

This result extends the known examples of CO spaces beyond successor ordinals, introducing a new, uncountable discrete example, and provides a clear structural picture that can serve as a foundation for further investigations into CO spaces arising from more general ordered or partially ordered structures.