We develop a logical approach to quantitative domain theory by introducing a notion of ideal which is appropriate in this setting. The definition is given for continuity spaces since in this way our theory includes ordinary domain theory, metric domains, ultrametric domains and other examples, such as probabilistic domains and structure spaces, which may be useful in programming language semantics. We develop the basic properties of quantitative ideals and use them to define directed completeness and a general version of the Scott topology. Moreover, we show that every continuity space has an ideal completion and that the Scott topology on a directed complete continuity space can be used to characterize Scott-continuous maps. We also show that directed completeness via quantitative ideals agrees with completeness via forward Cauchy nets.
Date received: January 1, 1996.