In mathematics and theoretical computer science the Lawson topology, named after J. D. Lawson, is a topology on partially ordered set

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

s used in the study of domain theory

Domain theory

Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...

. The lower topology on a poset P is generated by the subbasis consisting of all complements of principal filters

Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...

 on P. The Lawson topology on P is the smallest common refinement of the lower topology and the Scott topology on P.

Properties

• If P is a complete upper semilattice
  Semilattice
  In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
  
  , the Lawson topology on P is always a complete T₁ topology.

External links

• "How Do Domains Model Topologies?," Pawel Waszkiewicz, Electronic Notes in Theoretical Computer Science 83 (2004)