Inverse topological systems and compactness in abstract model theory

Journal of Symbolic Logic 51 (3):785-794 (1986)
Given an abstract logic L = L(Q i ) i ∈ I 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 $S_T = \{(S_\tau, \pi^\tau_\sigma)\mid\sigma, \tau \in T, \sigma \subset \tau\}$ is an inverse topological system whose bonding mappings π τ σ are naturally determined by the reduct operation on structures. We relate the compactness of L to the topological properties of S T . For example, if I is countable then L is compact iff for every τ each clopen subset of S τ is of finite type and S τ is homeomorphic to $\underset{lim}S_T$ , where T is the set of finite subtypes of τ. We finally apply our results to concrete logics
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DOI 10.2307/2274032
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Xavier Caicedo (1993). Compactness and Normality in Abstract Logics. Annals of Pure and Applied Logic 59 (1):33-43.

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