RETRACTED: Semantic Domains for Combining Probability and Non-Determinism

A domain-theoretic Banach–Alaoglu theorem

“…Then a d-cone is an R + -semimodule in the category of dcpos, where R + is endowed with the usual ordering with least element 0, making it a continuous lattice; an ordered cone is an R + -semimodule in the category of posets, endowing R + with the usual ordering; and a topological cone is an R + -semimodule in the category of topological spaces, endowing R + with the upper topology. Our definition of a d-cone differs inessentially from that in [20] where the infinite element is avoided.…”

“…So the force of the conclusion is more the local stable compactness of C * . The d-cone V(Ω) used in the counterexample above satisfies the conditions of the theorem (see [20], Chapter 2.2) and so also provides an example where the open-lower and the weak * -lower topologies disagree: {F ∈ V(Ω) * | F (η ∞ ) ≥ 1} is closed in the latter but not the former.…”

“…Their properties are treated in detail in Chapter 2 of [20]; we note here a 'Riesz Representation Theorem,' that Λ :…”

“…The weak * -lower topology is always a separating dual topology for the weak * -upper topology (see [20], Definition VI-6.17), and its specialisation ordering is the pointwise one.…”

“…In the latter, 'notions of positivity and order play a key rôle,' as do lower semicontinuity and (so) T 0 spaces. The present paper contributes a domain-theoretic analogue of the classical Banach-Alaoglu theorem for continuous d-cones, that is, domains endowed with a compatible cone structure [20].…”