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Reviewed by Marek Balcerzak (Łód´z) References

By Harold R. (-txt) Lutzer, J. Aarts and D. Lutzer


The article contains important results on domains and Scott domains, which are special kinds of partially ordered sets endowed with a natural topology called the Scott topology. Let max(P) denote the set of maximal elements in a given domain (P, ⊑). Theorem 1.1 describes several properties of a space X ⊆ max(P) provided that X is a Gδ-subset of P with the Scott topology and (P, ⊑) is a Scott domain. In particular, it follows that [0, ω1) cannot be max(P) for any Scott domain having max(P) as a Gδ-subset. This corrects a claim stated in the literature. A domain can carry a special kind of function µ from P to [0, ∞) called a measurement. The authors use a space constructed by D. K. Burke [General Topology and Appl. 2 (1972), 287–291; MR0319156 (47 #7702)] to give a negative answer to the so-called MMR Question on measurements with kernels equal to max(P). The next result, Theorem 1.3, is a counterpart of Theorem 1.1 for domains that are not necessarily Scott domains. An example is given of a space to which Theorem 1.3 is applicable but Theorem 1.1 does not work. Another example (using Q-sets) shows that the kernel of measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space. Finally, some open questions are formulated

Year: 2013
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