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A compact Hausdorff space X is called a CO space, if every closed subset of Xis homeomorphic to an open subset of X. Every successor ordinal with its ordertopology is a CO space. We find an explicit characterization of the class K ofCO spaces which are a continuous image of a Dedkind complete totally orderedset. (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 ofvery simple spaces. Every summand has either form: (1) mu + 1 + nu^*, where muand nu are cardinals, and nu^* is the reverse order of nu; or (2) the summandis the 1-point-compactification of a discrete space with cardinality aleph_1.

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