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Order-Compactifications of Totally Ordered Spaces: Revisited

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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 13:56–65, 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.

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  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003-8001, USA

    Guram Bezhanishvili & Patrick J. Morandi

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  1. Guram Bezhanishvili
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  2. Patrick J. Morandi
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Correspondence to Patrick J. Morandi.

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Bezhanishvili, G., Morandi, P.J. Order-Compactifications of Totally Ordered Spaces: Revisited. Order 28, 577–592 (2011). https://doi.org/10.1007/s11083-010-9193-x

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