Scott continuity
Appearance
In
supremum in P, its image
has a supremum in Q, and that supremum is the image of the supremum of D, i.e. , where is the directed join.[1] When is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.[2]
A subset O of a partially ordered set P is called Scott-open if it is an
continuous with respect to the Scott topology.[1]
The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.[3]
Scott-continuous functions are used in the study of models for
lambda calculi[3] and the denotational semantics
of computer programs.
Properties
A Scott-continuous function is always
monotonic
, meaning that if for , then .
A subset of a directed complete partial order is
lower set and closed under suprema of directed subsets.[4]
A
inclusion.[5]
For any Kolmogorov space, the topology induces an order relation on that space, the
open neighbourhood of x is also an open neighbourhood of y. The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be sober: the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.[4]
Examples
The open sets in a given topological space when ordered by
open neighbourhoods of X is open with respect to the Scott topology.[5]
For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.[6]
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.[7]
See also
Footnotes
- ^ ISBN 978-0-521-36062-3.
- ^ Scott topology at the nLab
- ^ Lawvere, Bill(ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.
- ^ ISBN 978-0-19-853762-5.
- ^ S2CID 6774320. Retrieved October 8, 2010.
- ISBN 978-0-444-87508-2. (See theorems 1.2.13, 1.2.14)
- ISBN 0-85362-161-6