In this paper we return to the continuous normalized valuations Μ on the systems of open sets and introduce notions of co-continuity ({U _i , i ∃ I} is a filtered system of open sets ⟹ Μ(Int(∩ _i∃I U_i)) = inf _i∃I Μ(U_i)) and strong non-degeneracy (U ⊂ V are open sets ⟹ Μ(U) < Μ(V)) 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 Μ on the system of Scott open subsets of A, we construct a continuous partial metric on A yielding the Scott topology as u(x, y)=Μ(A (C _x ∩ C_y)) − Μ(I_x ∩ I_y), where C _x = {y ∃ A ¦ y ⊑ x } and I _x = {y ∃ A ¦ {x, y } is unbounded }. This construction covers important cases based on the real line and allows to obtain an induced metric on Total(A) without the unpleasant restrictions known from earlier work.