Scott topology

The Scott topology , named after Dana Scott , is a topology that results from the partial ordering on a semi-ordered set. Among other things, it plays a role in theoretical computer science .

definition

It is a lot with partial order . A subset is called Scott-closed , if ${\displaystyle (A,\sqsubseteq )}$ ${\displaystyle U\subseteq A}$

• ${\displaystyle U}$with respect to a subset , that means with each element also each with respect to the partial order contains smaller ones, and ${\displaystyle \sqsubseteq }$
• all directed that in a supremum have is . ${\displaystyle M\subseteq U}$ ${\displaystyle (A,\sqsubseteq )}$ ${\displaystyle \bigsqcup M}$ ${\displaystyle \bigsqcup M\in U}$

Thus defined Scott-closed sets are exactly the closed sets of Scott topology on . ${\displaystyle (A,\sqsubseteq )}$

properties

In the following, let and be semi-ordered sets and they are endowed with the respective Scott topology. ${\displaystyle (A,\sqsubseteq _{A})}$ ${\displaystyle (B,\sqsubseteq _{B})}$

• If it is a continuous mapping, then is monotonic. ${\displaystyle f\colon A\to B}$ ${\displaystyle f}$
• A mapping is continuous if and only if directed suprema is given, i.e. H. for all directed with Supremum it is . ${\displaystyle f\colon A\to B}$ ${\displaystyle f}$ ${\displaystyle M\subseteq A}$ ${\displaystyle \bigsqcup M}$ ${\displaystyle \bigsqcup (f(M))=f(\bigsqcup M)}$

literature

S. Abramksy, A. Jung: Handbook of Logic in Computer Science . Vol. III. Oxford University Press, 1994, ISBN 0-19-853762-X , Domain theory ( bham.ac.uk [PDF]).

Web links

Scott topology , entry in the nLab . (English)

Individual evidence

1. ^ Dana Scott Continuous lattices , in Lawvere Toposes, Algebraic Geometry and Logic , Lecture Notes in Mathematics 274. Springer-Verlag 1972