Monads and realcompactness
A completely regular ordered topological space \((X,T,\leq)\) (in the sense of [\textit{L. Nachbin}, Topology and order (1965; Zbl 0131.379)]) is ordered real-compact if it is order isomorphic to a closed subspace of a product of copies of the usual ordered space \(\mathbb{R}\) of reals. The authors' main result is to give a ``quantifier free'' characterization of ordered realcompactness (which agrees with usual realcompactness when the order is discrete) in terms of A. Robinson-style nonstandard analysis. In particular, define a point \(\alpha\in{^*X}\) to be prenearstandard if \(^*f (\alpha)\) is finite in \(^*\mathbb{R}\) for every monotone nondecreasing continuous function \(f:X \to \mathbb{R}\). As usual, \(\alpha\) is nearstandard if there is some \(x \in X\) such that \(\alpha \in {^*U}\) for every open neighborhood \(U\) of \(x\). Clearly every nearstandard point of \(^*X\) is prenearstandard; the authors show that the converse holds if and only if \(X\) is ordered realcompact.
scientific article; zbMATH DE number 550033