In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, given two
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
s ''P'' and ''Q'', a
function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician
Dana Scott) if it preserves all
directed suprema. That is, for every
directed subset ''D'' of ''P'' with
supremum in ''P'', its
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
has a supremum in ''Q'', and that supremum is the image of the supremum of ''D'', i.e.
, where
is the directed join.
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.
A subset ''O'' of a partially ordered set ''P'' is called Scott-open if it is an
upper set and if it is inaccessible by directed joins, i.e. if all directed sets ''D'' with supremum in ''O'' have non-empty
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
with ''O''. The Scott-open subsets of a partially ordered set ''P'' form a
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on ''P'', the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is
continuous with respect to the Scott topology.
The Scott topology was first defined by Dana Scott for
complete lattices and later defined for arbitrary partially ordered sets.
Scott-continuous functions are used in the study of models for
lambda calculi and the
denotational semantics
In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called ''denotations'' ...
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
closed with respect to the Scott topology induced by the partial order if and only if it is a
lower set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
and closed under suprema of directed subsets.
A
directed complete partial order (dcpo) with the Scott topology is always a
Kolmogorov space (i.e., it satisfies the
T0 separation axiom).
However, a dcpo with the Scott topology is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
if and only if the order is trivial.
The Scott-open sets form a
complete lattice when ordered by
inclusion.
For any Kolmogorov space, the topology induces an order relation on that space, the
specialization order: if and only if every
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.
Examples
The open sets in a given topological space when ordered by
inclusion form a
lattice on which the Scott topology can be defined. A subset ''X'' of a topological space ''T'' is
compact with respect to the topology on ''T'' (in the sense that every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...
of ''X'' contains a
finite subcover of ''X'') if and only if the set of
open neighbourhoods of ''X'' is open with respect to the Scott topology.
For CPO, the
cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are
curry and
apply.
Nuel Belnap used Scott continuity to extend
logical connectives to a
four-valued logic.
[N. Belnap (1975) "How Computers Should Think", pages 30 to 56 in ''Contemporary Aspects of Philosophy'', Gilbert Ryle editor, Oriel Press ]
See also
*
Alexandrov topology
In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov top ...
*
Upper topology
Footnotes
References
* {{planetmath reference, urlname=ScottTopology, title=Scott Topology
Order theory
General topology
Domain theory