A compactification for convergence ordered spaces

D. C. Kent, G. D. Richardson
1984 Canadian mathematical bulletin  
Compactifications are constructed for convergence ordered spaces and topological ordered spaces with extension properties that resemble those of the Stone-Cech compactification. 0. Introduction. One of the authors [4] introduced a convergence space compactification with an extension property similar to that of the topological Stone-Cech compactification. We later showed in [2] that the compactification of [4] gives rise to a topological compactification with an interesting lifting property.
more » ... fting property. This work is concerned with convergence ordered spaces, a natural generalization of the topological ordered spaces of Nachbin [3]. By "convergence ordered space" we mean a partially ordered set with a convergence structure generated by filters which have bases of convex sets. A preliminary section gives a brief introduction to such spaces. In Section 2, a convergence ordered compactification is constructed for an arbitrary convergence ordered space by defining an appropriate partial order on a class of filters and using a "Wallman-type" construction similar to that of [4] . The extension properties of this compactification are examined in Section 3; in addition to generalizing the extension results of [4], conditions are found subject to which ours is the largest convergence ordered compactification. The last section applies the results of the preceding sections to obtain a topological ordered compactification with similar lifting properties. Choe and Park [1] have constructed a Wallman ordered compactification for the topological setting. It is shown, under certain assumptions, that our topological ordered compactification is larger than that by Choe and Park. 1. Preliminaries. Let (X, <) be a partially ordered set (or poset) equipped with a convergence structure. A convergence structure on X is a relation-> between the set F(X) of all filters on X and X which satisfies the following
doi:10.4153/cmb-1984-081-6 fatcat:bo3sax2xlnhilprap3dwqapzye