From there you just have to observe that the the open sets of a space X must correspond to the maps from X to the Sierpinski space. Indeed, they are excatly the continuous inverse images of the open point of the Sierpinski space. If the domain-theoretic meaning of continuity is to match the topological meaning of continuity, then open sets of domains ought to be in the same kind of correspondence with domain-continuous functions into T.

Well, so what is domain-continuity? It's monotonicity and preservation of ω-chain lubs. Suppose I've got a domain-continuous function f into T and I care about the inverse image U = f

^{-1}(⊤). Monotonicity says that if x ≤ y, then f(x) ≤ f(y). The only violations of this are when x ≤ y but f(x) = ⊤ and f(y) = ⊥. Thus the monotonicity condition on my inverse image is,

**if x ∈ U, and x ≤ y, then y ∈ U, too**. This is the part I often remember: Scott-open sets

*are*up-closed. They're not all the up-closed subsets of the ω-cpo, though. We didn't cover preservation of limits yet. If f is continuous then

_{i}) = sup f(a

_{i})

_{i}. That sup on the right-hand side is taking place inside T, though, so it's easy: we just return ⊤ iff there is some i such that f(a

_{i}) = ⊤. Since f(x) = ⊤ is the same thing as x ∈ U, this means the condition is

_{i}∈ U iff there exists i such that a

_{i}∈ U

**if sup a**.

_{i}∈ U, then there exists i such that a_{i}∈ U