A curious example involving ordered compactifications

Thomas A. Richmond


For a certain product X x Y where X is compact, connected, totally ordered space, we find that the semilattice K0 (X x Y) of ordered compactifications of X x Y is isomorphic to a collection of Galois connections and to a collection of functions F which determines a quasi-uniformity on an extended set X U {+∞}, from which the topology and order on X is easily recovered. It is well-known that each ordered compactification of an ordered space X x Y corresponds to a totally bounded quasi-uniformity on X x Y compatible with the topology  and order on X x Y, and thus K0 (X x Y) may be viewed as a collection of quasi-uniformities on X x Y. By the results here, these quasi-uniformities on X x Y determine a quasi-uniformity on the related space X U {+∞}.


Ordered topological space; Ordered compactification; Galois connection; Quasi-uniformity; F-poset

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Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt