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Strong Minimality in Continuous Logic

Recall that a set S is totally bounded if for every > 0, S has a finite -net. Recall further that if S is totally bounded in a complete metric space, then S is compact. We will say that a theory T is strongly minimal if for every M T , the type {x = x} is strongly minimal. That is, for everyM T , every zero set of a definable predicate is either totally bounded or its complement in M is. Recall that in classical logic, a theory is called strongly minimal if all of the definable sets in anyM T are finite or cofinite. So the above notion of strong minimality for continuous logic is analogous to that for classical logic (in that compactness is analogous to finiteness) but it is actually a little bit stronger, since in general, not every zero set is definable. A benefit of this notion of strong minimality is that the following fact from classical logic still holds: