Scott continuity

Source: Wikipedia, the free encyclopedia.

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

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

  1. ^ .
  2. ^ Scott topology at the nLab
  3. ^
    Lawvere, Bill
    (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.
  4. ^ .
  5. ^ . Retrieved October 8, 2010.
  6. . (See theorems 1.2.13, 1.2.14)

References