From: Dana Scott <dana.scott@cs.cmu.edu>
To: Michael Barr <barr@math.mcgill.ca>
Cc: Categories list <categories@mta.ca>
Subject: Re: Does this topology have a name?
Date: Sat, 1 Jan 2011 10:53:55 -0800 [thread overview]
Message-ID: <E1PZPSL-0001DI-8o@mlist.mta.ca> (raw)
In-Reply-To: <E1PZ4ag-00027i-Mu@mlist.mta.ca>
On Jan 1, 2011, at 6:59 AM, Michael Barr wrote:
> Let A be a model of a finitary equational theory and let X be the set of
> congruences on A. For a,b in A, let M(a,b) = {E} such that E is a
> congruence on A and aEb. Does this topology have a name? It turns out
> that this topology is coherent which means, among other things, that if we
> make the M(a,b) clopen, the result is a Stone space.
Consider the powerset space P(A x A) = 2^(A x A). The product topology
makes it a Stone space. This is elementary.
Now, the space X of congruences is defined by logical formulae with
only universal quantifiers and atomic formulae xEy for variables
ranging over A. That makes X a CLOSED subspace of P(A x A).
This is so easy, it hardly needs a name. And it works even if A has
infinitely many operations. That there is an equational theory in the
background seems neither here nor there to get a Stone space of congruences.
HAPPY NEW YEAR!
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next prev parent reply other threads:[~2011-01-01 18:53 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2011-01-01 14:59 Michael Barr
2011-01-01 18:53 ` Dana Scott [this message]
[not found] ` <9CF079C4-EA66-4563-9DB0-FB3942D47DD9@cs.cmu.edu>
2011-01-01 22:15 ` Michael Barr
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