Order-Compactifications of Totally Ordered Spaces: Revisited

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13: 56-65, 1975) and by Kent and Richmond (J Math Math Sci 11(4): 683-694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorcuk (Soviet Math Dokl 7: 1011-1014, 1966; Sib Math J 10: 124-132, 1969) and Kaufman (Colloq Math 17: 35-39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13: 56-65, 1975) and Kent and Richmond (J Math Math Sci 11(4): 683-694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.