The Scott topology is a topology that can be defined on any complete partial order (cpo).
Thus, the Scott topology is important for domain theory and the semantics of recursion. However, because the resultant space is a space, there is very little geometric intuition involved.
Let be a -cpo. We define to be Scott-open whenever
Let be a dcpo. We define to be Scott-open whenever
The nLab gives a more synthetic definition.
Let be the Sierpiński space, i.e. the cpo
Let be a cpo, and . The characteristic function is given by
We then define to be Scott-open whenever is monotone and continuous with respect to the two cpo's.
A topology can be equivalently described through its closed sets. We can then describe the Scott-topology through the Scott-closed sets.
We can recover the order from the topology, by defining
Plotkin, Gordon. 1983. ‘Domains (Pisa Notes)’. https://homepages.inf.ed.ac.uk/gdp/publications/Domains_a4.ps.
@techreport{plotkin_1983,
title = {Domains ({Pisa} {Notes})},
url = {https://homepages.inf.ed.ac.uk/gdp/publications/Domains_a4.ps},
author = {Plotkin, Gordon},
year = {1983},
}