I wrote to Barr as follows:
From: Dana Scott <dana.scott@cs.cmu.edu> Date: January 1, 2011 2:57:30 PM PST To: Michael Barr <barr@math.mcgill.ca> Subject: Re: categories: Does this topology have a name?
On Jan 1, 2011, at 2:15 PM, Michael Barr wrote:
Yes, the fact that when these sets are taken as clopens gives a Stone space is easy. But I want to know what to call the weaker topology in which you take these sets as a basis of opens.
Ah, I had thought you meant the clopen case. The weaker topology is (unfortunately) called the Scott topology, which can be given to any algebraic lattice. The congruences form an algebraic lattice inasmuch as they are closed under arbitrary intersections and directed unions. (Yes?) Details are in the book: Continuous Lattices and Domains by Gierz/Hofmann/Keimel/Lawson/ Mislove/Scott.
A little more detail: The opens in the lattice of congruences are determined by the "compacts" of this algebraic lattice. These are the finitely generated congruences. If F is one such, then the open it determines is {E | F subset E}. They form a basis for the "Scott" topology. A subbasis is given by the sets {E | aEb} indicated by Barr. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]