Topology and Stone Duality for Domains Uday S. Reddy May 8, 1995 1 Topology in a nutshell The basic ideas of topology are contained in the two definitions below. Definition 1 A(topological) space X = hX; ΩX i is a pair consisting of a set X (of points x; y; z) and a collection ΩX of subsets of X (open sets u; v; w) such that ΩX is closed under arbitrary union and finite intersection. S If s is a family of open sets, s is an open set. In particular, ? is an open set. If u1; : : : ; un is a family of open sets then u1 \···\ un is an open set. In particular, X is an open set. We use the terminology \u is a neighborhood of x" when u is an open set containing a point x. Definition 2 If X and Y are topological spaces, a function f : X ! Y is said to be continuous at x 2 X if, for every neighborhood v of f(x), there exists a neighborhood u of x such that f(u) ⊆ v. (By f(u), we mean the set f f(x): x 2 u g.) A continuous function is a function that is continuous at every x 2 X. Topological spaces and continuous functions form a category Top. For theoretical purposes however, it is often simpler to use the following criterion for continuous functions: Theorem 3 A function f : X ! Y is continuous iff, for every open set −1 v 2 ΩY , the inverse image f (v) is an open set in ΩX . Proof: \If:" If f −1(v) is an open set, take the requisite witness u to be f −1(v). \Only if:" For every x 2 f −1(v), f(x) 2 v. By definition, there exists a neighborhood ux of x (an open set) such that f(ux) ⊆ v. By generalization, S −1 x2f −1(v) ux is an open set whose direct image is included in v. So f (v) = S −1 x2f −1(v) ux, and we have that f (v) is an open set. From here on, we take this to be our working definition of continuous functions. 1 The main motivating example for topological spaces is the real line. Let R denote the set of real numbers. The \standard" topology on R consists of open sets that are unions of open intervals (a; b). Evidently, any union of open sets is an open set. The entire set R is an open set because it S is, for instance, the union x2R(x − , x + ) for some positive number . The intersection of open sets is an open set because the intersection of open intervals is an open interval. The definition of R as an open set suggests an equivalent criterion for open sets in the standard topology. A subset u is open iff, for every x 2 u, there exists an open interval (a; b) ⊆ u such that x 2 (a; b). This leads to the usual notion of a continuous function on the real line familiar from calculus: a function f : R ! R is continuous at x, iff, for every neighborhood v = (f(x)−, f(x)+) of f(x), there exists a neighborhood u = (x−δ; x+δ) such that f(u) ⊆ v. Definition 4 A basis for a space X is a family of subsets σ such that a subset u ⊆ X is open iff it is expressible as S s for some s ⊆ σ. The elements of σ are called \basic" open sets (relative to σ). For the real line, the set of open intervals (a; b) forms a basis. For that matter, the set of rational open intervals (a; b) forms a basis too. The definition of continuity can now be refined. A function f : X ! Y is continuous if, for every basic open set v of Y , f −1(v) is an open set in X. The example of real line gives a \geometric" intuition to the subject of topology. For our purposes, it is also useful to keep in mind a \logical" intuition. Regard the open sets as (perhaps finitary) pieces of information about their constituents. For instance, the open interval (x − δ; x + δ) says that each of its elements is within δ distance from x. Union of open sets then corresponds to infinitary disjunction of information. The empty set denotes the \impossibility." Intersection corresponds to finitary conjunc- tion of information. Continuity of functions says that each (finitary) piece of information about f(x) can be obtained from some (finitary) piece of information about x. These are, indeed, the basic intuitions underlying domain-theoretic semantics for programming languages. 2 Domains as topologies Consider Scott domains (bounded-complete algebraic cpo's). We write D0 for the set of finite elements (compact elements) of D. Every x 2 D can be written as x = F(#x) where #x is the downward-closure f y : y v x g. Since D is algebraic, we can in fact write x = F(#0x) where #0x = (#x) \ D0. We have x v y iff #0fxg ⊆ #0fyg. A Scott-continuous function f : D ! E preserves directed sups. There is an equivalent criterion: 2 Theorem 5 A function f : D ! E is Scott-continuous iff, whenever y0 v f(x) is a finite element, there exists a finite approximation x0 v x such that y0 v f(x0). Proof: \Only if:" Suppose y0 v f(x) is a finite element. Consider the 0 F F directed set d = # x. Since x = d, we have f(x) = f(d). But y0 is a finite element, so it is bounded by an element in f(d), say f(x0) where x0 2 d. We have that y0 v f(x0), x0 a finite approximation of x. \If:" Suppose d ⊆ D0 is a directed set with Fd = x. Without loss of generality, assume that d is down-closed. Then, for every finite y0 v f(x), F we have y0 v f(x0) v f(d) (where x0 is an appropriate finite element in d). This shows f(x) v Ff(d). The reverse inclusion Ff(d) v f(x) follows by monotinicity. Hence, f(x) = Ff(d). If x is a finite element, there is no reason for f(x) to be a finite element. But, if f(x) is finite, then all its information is already contained in f(x0) for some finite approximation x0 v x. Let us restate the criterion of the theorem in terms of upper sets. \y0 v f(x)" is equivalent to f(x) 2 "fy0g, and \x0 v x" is equivalent to x 2 "fx0g. So, f continuous means that whenever f(x) is in a compact upper set "fy0g, there exists a compact upper set "fx0g containing x such that f("fx0g) ⊆ "fy0g. Thus, we may regard compact compact upper sets "fx0g and unions thereof, as open sets. Definition 6 If D is a Scott domain, a Scott-open set in D is an upper set "a of a subset a ⊆ D0. The collection of Scott-open sets forms a topology on D, called the Scott topology. To see that this is a topology, note that the union of upper sets Ss is an upper set and the minimal elements are still finite. The intersection of upper sets "a \"b is "f x t y : x 2 a; y 2 b g if x t y exists and ? otherwise. Corollary 7 A function f : D ! E between Scott domains is Scott- continuous iff it is (topologically) continuous with D and E regarded as Scott topologies. We may now import geometric as well as logical intuitions to Scott domains. Geometrically speaking, we regard every Scott-open set "a as a neighborhood for every element x 2 "a. A \smaller" neighborhood "b of x satisfies the \upper" order a vU b () 8y0 2 b: 9x0 2 a: x0 v y0, i.e., every point of b is above some point of a. Continuity says that we can get arbitrarily close to f(x) by getting arbitrarily close to x. Logically speaking, the finite elements of D represent finitary pieces of information about the elements of D. The finite element ?D says \nothing" ("f?Dg = D) while x0 t y0 gives the conjunction of information given by x0 and y0 ("fx0 t y0g = "fx0g \ "fy0g). If there is no upper bound for x0 and 3 y0, then their information is \inconsistent" together, i.e., "fx0g \ "fy0g = S ?. Disjunction of information is given by union ("a = x2a "fxg). All this is certainly implicit in our usual understanding of Scott domains. The topological formulation makes it explicit. We will introduce a bit of notation. If u and v have a non-empty intersection, we write u ^_ v to mean that they are consistent. If the intersection is empty, we write u ^ v. (They are inconsistent.) If they are allowed to be equal as well as be inconsistent, we write u _^ v. 3 Topological notions The basic concept of a topological space is much too general (in the same vein as the notion of cpo being too general). A variety of \separation axioms" are often used to characterize the spaces of interest. A Hausdorff space or T2-space is defined by the axiom: whenever x and y are distinct points, there exist disjoint neighborhoods u and v for the two points respectively. One writes x 6= y =) 9u; v 2 ΩX : u ^ v ^ x 2 u ^ y 2 v or _ [8u; v 2 ΩX : x 2 u ^ y 2 v =) u ^ v] =) x = y So, in a Hausdorff space, all points are \independent" and do not impinge on each other.