Continuous Functions on Totally Ordered Spaces That Are Compact in Their Order Topologies

AbstractThe totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes. The Banach space of continuous functions on the lexicographic product of compact totally ordered spaces, whose spaces of continuous functions have locally uniformly convex norms, has an equivalent locally uniformly convex norm if; and only if, the product is countable. The Banach space of continuous functions on a compact totally ordered space always has an equivalent Kadec norm. Those compact totally ordered spaces, for which the Banach space of continuous functions has an equivalent locally uniformly convex norm, are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces. Some examples are discussed in detail