FOM: A question about model theory

[Andrej Bauer]

John Goodrick <goodrick at math.berkeley.edu> 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?

According to P.T. Johnstone "Stone Spaces", II.4.2., page 69:

    ``Theorem: the following are equivalent
    (i)   X is compact, Hausdorff and totally disconnected (the only connected
          sets are singletons).
    (ii)  X is compact and totally separated (any two points are separated
          by a clopen).
    (iii) X is compact, T_0 and zero-dimensional (clopens form a base).
    (iv)  X is Hausdorff and coherent (sober and the compact
          open subsets are closed under finite intersection and form a base).

    Proof: [omitted]

    A space satisfying the conditions of the Theorem will be called
    a _Stone space_.''

So, it seems like the answer to your question is "yes" by definition.
(Of course, Johnstone also proves that the category of Stone spaces is
the dual of the category of Boolean algebras.)
 
> 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?

Here I only remark that

    X is first-countable, totally disconnected, compact, Hausdorff

implies

          X is a continuous retract of the Cantor space 2^N

But I don't see how this might be helpful, since the obvious
conclusion drawn from that is that by duality, the countable atomless
Boolean algebra is a retract of the Boolean algebra corresponding to
X, which seems to be irrelevant for your question.

Andrej Bauer
Institute of Mathematics, Physics, and Mechanics
Ljubljana, Slovenia
http://andrej.com/