Ordered spaces and complete uniformizability

The order structure on M defined in Chaps. 3 and 4 determines not just a topology but also a family of uniformities on M, and a uniform structure admits a notion of completeness. The completed space can carry mathematical structures (such as differentiable or analytic structures) that cannot be carried by ordered spaces like ℚ². These structures were first elaborated in the context of metric spaces, but ordered spaces may be uniformizable without being metrizable. In this chapter, we shall review the completion of uniformities determined by the order structure, define a notion of order uniformity and its completion, and analyse the problem of extending the order to the completed space. This will lead us to a concept which we shall call order completion (and which will be slightly different from uniform completion), and it will be clear from the definition that every ordered space has an order completion.