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Intrinsic approach spaces on domains. (English) Zbl 1231.06012

Summary: The paper is a contribution to quantifiability of domains. We show that every domain \(X\), regardless of cardinality conditions for a domain bases, is quantifiable in the sense that there exists an approach structure on \(X\) [R. Lowen, Approach spaces: the missing link in the topology-uniformity-metric triad. Oxford: Clarendon Press (1997; Zbl 0891.54001)] defined by means of a gauge of quasi-metrics, inducing the Scott topology. We get weightability for free and in the case of an algebraic domain satisfying the Lawson condition [J. Lawson, Math. Struct. Comput. Sci. 7, No. 5, 543–555 (1997; Zbl 0985.54025)] a quantifying approach space can be obtained with a weight satisfying the kernel condition.

MSC:

06B35 Continuous lattices and posets, applications
06F30 Ordered topological structures
54A05 Topological spaces and generalizations (closure spaces, etc.)
54B30 Categorical methods in general topology
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