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