Re; Does this topology have a name?

[Dana Scott <dana.scott@cs.cmu.edu>]

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.




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