On Vervaat's sup vague topology (Q909288)

Let \({\mathcal F}(S,I)\) be the collection of upper semicontinuous functions from S into I, where S is a locally quasicompact topological space and I is a compact interval on the extended real line. It is known that Vervaat's sup vague topology on \({\mathcal F}(S,I)\) is both compact and Hausdorff. We give a nonstandard proof based on Robinson's characterization of compactness. Its main step is a characterization of the standard part map.