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A characterization of partial metrizability: Domains are quantifiable. (English) Zbl 1043.54011
Following {\it S. G. Matthews} [Ann. N.Y. Acad. Sci. 728, 183--197 (1994; Zbl 0911.54025)] a quasi-metric space $(X, d)$ is called weightable if there exists a mapping $w: X\to [0,\infty)$ such that $d(x, y) +w(x)= d(y, x)+w(y)$ for each pair $x,y\in X$. It is known that every weightable quasi-metric space is quasi-developable. However, it is an open problem to characterize those quasi-uniform spaces $(X, {\Cal U})$ for which there exists a weightable quasi-metric $d$ an $X$ such that $(X,{\Cal U})= (X,{\Cal U}_d)$. Therefore the following theorem is interesting. Theorem 1. Let $(X, {\Cal U})$ be a quasi-uniform join semilattice, i.e., a quasi-uniform space such that (i) ($X,\le_{\Cal U})$ is a partially ordered set such that for each pair $x,y\in X$ there exists a supremum $x\vee y$, where $\le_{\Cal U}= \bigcap\{U\mid U\in{\Cal U}\}$, and (ii) the mapping $(x, y)\mapsto x\vee y$ is quasi-uniformly continuous. Then there exists a weightable invariant quasi-metric $d$ an $X$ such that $(X, {\Cal U})= (X, {\Cal U}_d)$ if and only if there exists a $Q$-join co-valuation on $(X,{\Cal U})$. It is shown that this theorem is strong enough to solve a major problem from domain theory. Theorem 2. Every domain (i.e., every directed complete partially ordered set $(P, \sqsubseteq)$ with a countable basis) is quantifiable (i.e., there exists a weightable quasi-metric $d$ an $P$ such that the topology induced by $d$ coincides with the Scott-topology of $P$ and the associated preorder $\le_d$ coincides with $\sqsubseteq$).

54E15Uniform structures and generalizations
54E35Metric spaces, metrizability
06B35Continuous lattices and posets, applications
Full Text: DOI
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