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@MISC{Heckmann97quantitativesemantics,,
    author = {Reinhold Heckmann and Michael Huth},
    title = {Quantitative Semantics, Topology, and Possibility Measures},
    year = {1997}
}

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Abstract:

Given a topological space X and a complete lattice L, we study the space of L- predicates F L (X) = [X ! L op ] op , continuous maps from X to L op in its Scotttopology. It yields a functor F L (\Delta) from TOP-L, a full subcategory of TOP subsuming continuous domains, to SUP, the category of complete sup-lattices and maps preserving suprema. Elements of F 2 (X) are continuous predicates (= closed sets), and elements of F [0;1] (X) may be viewed as probabilistic predicates. Alternatively, one may consider the complete sup-lattice P L (X) = O(X)\Gamma ffi L of maps ¯: O(X) ! L preserving suprema (= possibility measures), which results in another functor P L (\Delta) from TOP to SUP. We show that these functors are equivalent for two restrictions. First, we leave SUP unchanged and restrict TOP-L to CONT, the category of continuous domains in their Scott-topology; second, we fix TOP but restrict L to co-continuous lattices. Conversely, if F L (X) and P L (X) are isomorphic f...

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