Order-Compactifications of Totally Ordered Spaces: Revisited

Abstract

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 Fedorčuk (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][57][58][59][60][61][62][63][64][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.

Ultrafilter Extensions of Models

“…the interval topology). Arbitrary ordered compactification of X has either one or two elements filling each proper gap of X, so the family of all ordered compactifications of X is isomorphic to the powerset of the set of its proper gaps (see [4]- [6] and recent review [7]). For more on linearly ordered sets we refer the reader to [3].…”