  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.
