In mathematics and theoretical computer science the Lawson topology, named after Jimmie D. Lawson, is a
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
on
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s used in the study of
domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer ...
. The lower topology on a
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
''P'' is generated by the subbasis consisting of all complements of principal
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on ''P''. The Lawson topology on ''P'' is the smallest common refinement of the lower topology and the
Scott topology
Scott may refer to:
Places Canada
* Scott, Quebec, municipality in the Nouvelle-Beauce regional municipality in Quebec
* Scott, Saskatchewan, a town in the Rural Municipality of Tramping Lake No. 380
* Rural Municipality of Scott No. 98, Saskat ...
on ''P''.
Properties
* If ''P'' is a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
upper
semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a mee ...
, the Lawson topology on ''P'' is always a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
T1 topology.
See also
*
Formal ball
In topology, a formal ball is an extension of the notion of ball to allow unbounded and negative radius. The concept of formal ball was introduced by Weihrauch and Schreiber in 1981 and the negative radius case (the generalized formal ball) by Ts ...
References
* G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove,
D. S. Scott
Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, ...
(2003), ''Continuous Lattices and Domains'', Encyclopedia of Mathematics and its Applications, Cambridge University Press.
External links
How Do Domains Model Topologies?" Paweł Waszkiewicz, Electronic Notes in Theoretical Computer Science 83 (2004)
{{topology-stub
Domain theory
General topology
Order theory