Inverse Topological Systems and Compactness in Abstract Model Theory

Given an abstract logic , generated by a set of quantifiers Q i , one can construct for each type τ a topological space S τ , exactly as one constructs the Stone space for τ in first-order logic. Letting T be an arbitrary directed set of types, the set is an inverse topological system whose bonding mappings are naturally determined by the reduct operation on structures. We relate the compactness of to the topological properties of S T . For example, if I is countable then is compact iff for every τ each clopen subset of S τ is of finite type and S τ , is homeomorphic to lim S T , where T is the set of finite subtypes of τ . We finally apply our results to concrete logics.