Talking about o-minimal theories in topological terms.
- Is there a standard way to talk about whether a given structure is o-minimal or not in topological terms?
I read about o-minimal theories the other day in A Shorter Model Theory in an exercise on page 29. The definition is compatible with the model-theoretic definition given in the Wikipedia article.
As stated in the Wikipedia article, a theory is o-minimal if and only if all of its models are o-minimal structures.
The first thing that I noticed is the definition appears to have a topological look and feel to it.
I tried to rephrase this definition in topological terms by making the definable sets clopen and generating a topology. But this construction fails for countable models where every singleton is nameable and reduces to a trivial topology where every subset is open and closed.