Jason (jcreed) wrote,
Jason
jcreed

For those of y'all that find yourselves like me constantly forgetting the definition of the Scott topology for ω-cpos, here's an easy way to recover it from a small piece of information: The small piece of information is that the smallest nontrivial ω-cpo, that is, the two-element ω-cpo T = {⊥ ≤ ⊤} has the smallest nontrivial topology, that is, Sierpinski topology, with ⊤ being the one open point.

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
f(sup ai) = sup f(ai)
for any ω-chain ai. 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(ai) = ⊤. Since f(x) = ⊤ is the same thing as x ∈ U, this means the condition is
sup ai ∈ U iff there exists i such that ai ∈ U
This is a condition I often forget, although it makes some sense: every limit element of an open set is in there for a 'finite' reason. Since we already have monotonicity, we only need one direction of this: if sup ai ∈ U, then there exists i such that ai ∈ U.
Tags: domain theory, math
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