*(English)*Zbl 0945.06006

Summary: The existence of deep connections between partial metrics and valuations is well known in domain theory. However, the treatment of non-algebraic continuous Scott domains has been not quite satisfactory so far.

In this paper we return to the continuous normalized valuations $\mu $ on the systems of open sets and introduce notions of co-continuity ($\{{U}_{i},i\in I\}$ is a filtered system of open sets $\Rightarrow \mu \left(\text{Int}\left({\bigcap}_{i\in I}{U}_{i}\right)\right)={inf}_{i\in I}\mu \left({U}_{i}\right)$) and strong non-degeneracy ($U\subset V$ are open sets $\Rightarrow \mu \left(U\right)<\mu \left(V\right)$) for such valuations. We call the resulting class of valuations CC-valuations. The first central result of this paper is a construction of CC-valuations for Scott topologies on all continuous dcpo’s with countable bases. This is a surprising result because neither co-continuous nor strongly non-degenerate valuations are usually possible for ordinary Hausdorff topologies.

Another central result is a new construction of partial metrics. Given a continuous Scott domain $A$ and a CC-valuation $\mu $ on the system of Scott open subsets of $A$, we construct a continuous partial metric on $A$ which yields the Scott topology as $u(x,y)=\mu (A\setminus ({C}_{x}\cap {C}_{y}))-\mu ({I}_{x}\cap {I}_{y})$, where ${C}_{x}=\{y\in A\mid y\u2291x\}$ and ${I}_{x}=\{y\in A\mid \{x,y\}$ is unbounded}. This construction covers important cases based on the real line and makes it possible to obtain an induced metric on $\text{Total}\left(A\right)$ without the unpleasant restrictions known from earlier work.