Abstract
This chapter describes the construction of the Stone–Cech compactification of a pospace in terms of real-valued functions on X. A pospace is a partially ordered topological space. Both Nachbin's and Priestley's definitions of complete regularity have their own interest. These definitions are reformulated in terms of zero-sets of functions. The chapter presents a theorem that states that a pospace X is completely regular (C.R.) if and only if it is isomorphic to a subpospace (resp. order subpospace) of some power of I. A pospace is compact Hausdorff if and only if it is isomorphic to some closed subspace of some power of I. In particular, any compact Hausdorff pospace is strongly completely regular (S.C.R.). Any compact Hausdorff pospace is C.R. Compact pospaces might be defined as locally convex pospaces in which any cover by convex open sets admits a finite subcover. In the case the pospace is Hausdorff, it is equivalent, however, to demand that the pospace be compact as a topological space.
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