Inverse topological systems and compactness in abstract model theory

Abstract

AbstractGiven an abstract logic , generated by a set of quantifiers Qi, one can construct for each type τ a topological space Sτ, 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 ST. For example, if I is countable then is compact iff for every τ each clopen subset of Sτ is of finite type and Sτ, is homeomorphic to limST, where T is the set of finite subtypes of τ. We finally apply our results to concrete logics.