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! [For admin and other information see: http://www.mta.ca/~cat-dist/ ]