Scott topology
Antonino Salibra
November 26, 2015
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0.1 Scott Topology
0.1.1 Posets and lattices
Let (X, ≤) be a poset and Y ⊆ X. An element a ∈ X is an upper bound of Y if a ≥ y for every y ∈ Y . An element a ∈ X is an lower bound of Y if a ≤ y for every y ∈ Y .
The least upper bound lub(Y ) of Y , when it exists, is un upper bound, which is minimum among the upper bounds: a upper bound of Y implies lub(Y ) ≤ a.
The greatest lower bound glb(Y ) of Y , when it exists, is un lower bound, which is maximum among the lower bounds: a lower bound of Y implies lub(Y ) ≥ a.
Figure 1: lub
A poset X is a lattice if, for all a, b ∈ X, lub({a, b}) and glb({a, b}) exist.
We denote by a ∨ b = lub({a, b}) and a ∧ b = glb({a, b}). In the power set of a set A, ∨ is the union of sets and ∧ is the intersection of sets. Every chain is a lattice (for example, the real line), where a ∨ b = max{a, b} and a ∧ b = min{a, b}.
A lattice is bounded if it has top element 1 and bottom element 0.
A bounded lattice is a Boolean if the lattice is distributive, that is x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), and every element a of L has a unique complement ¬a such that a ∧ ¬a = 0 and a ∨ ¬a = 1.
0.1.2 Scott
A subset Y of the plane is closed if it contains all the limits of the Cauchy sequences included within Y .
Definition 0.1.1. Let X be a poset. A downset C ⊆ X is Scott closed if every chain of C has lub in X if this lub exists. An upset O ⊆ X is Scott open if, for every chain I of X, if lub(I) exists and lub(I) ∈ O, then I ∩ O 6= ∅.
0.1. SCOTT TOPOLOGY 3
Figure 2: Lattices
Then, C is Scott closed if it contains all the existing limits (as lub) of chains included in C.
The Scott topology will be denoted by σX.
Example 1. (Partial Functions): Let f be a partial function. Then f ↑ is Scott open if, and only if, the (domain of the) function f is finite. It is easy to show that the upsets f ↑ with f finite generate the Scott topology in the cpo of partial functions from N into N.
(Power Set): Let B be a subset of a set A. Then B ↑ is Scott open if, and only if, the set B is finite. It is easy to show that the upsets B ↑ with B finite generate the Scott topology in the powerset of A.
(Real Line) Every subset of reals is a chain . Is an open interval (a, b) Scott open? No, because it is not an upset. Then Scott open sets are the intervals (a, ∞). To understand the difference with the usual Euclidean topology, consider an increasing sequence (x)n and a real a. If xi ≤ a for all i, then xi → a w.r.t. Scott topology iff xi → a w.r.t. Euclidean topology.
Lemma 0.1.1. Let X, Y be posets.
• A function f : X → Y is continuous w.r.t Alexandrov topology iff f is monotone, that is, x ≤ y implies f (x) ≤ f (y).
• A function f : X → Y is Scott continuous iff, for any increasing sequence (an), f (lub (an)) = lub f (an).
A fixpoint of a function f : A → B is an element a ∈ A such that f (a) = a.
A poset is a complete poset if every increasing chain has a least upper bound.
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Example 2.
