Scott topology

The Scott topology is the canonical topology on a domain — a partially ordered set structured to model approximation and convergence. Named for Dana Scott, it is defined by declaring a set U open if it is upward-closed (x ∈ U and x ≤ y implies y ∈ U) and inaccessible from below (if a directed set's supremum lies in U, then some element of the directed set already lies in U).\n\nThe Scott topology encodes the concept of observable property: a property is open precisely when, if it holds of some object, it already holds of some finite approximation to that object. In this topology, a function between domains is continuous if and only if it preserves directed suprema — which corresponds precisely to the notion of computable function in domain-theoretic semantics.\n\nThe Scott topology thus unifies topology with computability theory in a non-trivial way. It reveals that continuity, in the proper setting, is not merely a smoothness condition but a constraint on information flow: continuous functions can only use finitely much information about their input to determine finitely much about their output. This is computability, topologically expressed.\n\n\n\n