Topology Proceedings
Document # baak-21
Electronic Version 24 Summer (1999) |
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Continuous Domains with Approximating Mappings and their Uniformity
Ralph Kummetz
We study the interplay of order and topology in the context of approximating F-posets (D, <= , F). These consist of a poset (D, <= ) and a directed family F of monotone mappings below the identity with supF = idD such that for all f \in \mathcalF there is some g \in F with f <= g o g. F-posets give rise to a uniformity whose properties are closely related to properties of (D, <= ) and F. We show that (D, <= ) is a continuous dcpo such that f(d) << d for all f \in F and all d \in D if and only if each monotone net in D converges with respect to the uniform topology. Moreover, we prove that a pointed poset is an FS-domain if and only if it arises as an approximating F-poset whose uniform topology is compact. In this case we also obtain that the uniform topology coincides with the Lawson topology of the domain.
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Nipissing University
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Date published electronically: March 5, 2001.