FOM: A question about model theory

[Stephen G Simpson]

John Goodrick writes:
 > A quick question for the model theorists:
 > 
 > 1. Is any totally disconnected compact Hausdorff space homeomorphic to a
 > Stone space?  If not, do you have a counterexample?
 > 
 > 2. Is any first-countable, totally disconnected, compact Hausdorff space
 > homeomorphic to the Stone space of a countable set in a model with a
 > countable language?  Counterexample?

Since you are posing this as a question for model theorists, I am
going to assume you mean Stone spaces of the kind that come up in a
certain style of model theory.  Namely, by a Stone space you mean the
space S_1(A) of complete 1-types over a set A in a model of a complete
theory in the predicate calculus with identity.

The answer to 1 is yes.  The answer to 2 is yes if we replace first
countable by second countable.  In both results, we may take A to be
the empty set.

It is well known and not hard to see (though I don't have a reference
handy) that any totally disconnected compact Hausdorff space X is
homeomorphic to the space of completions of a theory T in the
propositional calculus.  (This is the Stone space of the Lindenbaum
sentence algebra of T.)  If X is second countable, T may be taken to
be countable.

Now, it is easy to write down a complete theory T' in the predicate
calculus with identity, corresponding to T.  For each propositional
atom in T there is a corresponding unary predicate in T'.  The space
of completions of T is homeomorphic to the space of complete 1-types
over (the empty set in any model of) T'.

-- Steve