FOM: A question about model theory

[Klaas Pieter Hart]

On Thu, 11 Jul 2002, John Goodrick wrote:

> A quick question for the model theorists:

Why not topologists?

> 1. Is any totally disconnected compact Hausdorff space homeomorphic to a
> Stone space?  If not, do you have a counterexample?

Yes, this is Stone's duality theorem; the space is the Stone space of its
Boolean algebra of closed-and-open sets.

> 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?

No, counterexample: the lexicographically ordered set [0,1]\times{0,1}
is, in the order topology, compact first-countable and totally
disconnected but(!) it does not have a countable base for its topology
(this is Alexandroff's double arrow space).
Furthermore: a compact totally disconnected space is the Stone space of
countable Boolean algebra iff it is second-countable.

KP

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