* Re; Does this topology have a name?
[not found] <7B51C5A3-329E-4BA4-A0E5-35A4F7E309E0@cs.cmu.edu>
@ 2011-01-02 18:30 ` Dana Scott
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From: Dana Scott @ 2011-01-02 18:30 UTC (permalink / raw)
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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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2011-01-02 18:30 ` Re; Does this topology have a name? Dana Scott
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