Compactness and Normality in Abstract Logics

Caicedo, X., Compactness and normality in abstract logics, Annals of Pure and Applied Logic 59 (1993) 33-43. We generalize a theorem of Mundici relating compactness of a regular logic L to a strong form of normality of the associated spaces of models. Moreover, it is shown that compactness is in fact equivalent to ordinary normality of the model spaces when L has uniform reduction for infinite disjoint sums of structures. Some applications follow. For example, a countably generated logic is countably compact if and only if every clopen class in the model spaces is elementary. The model spaces of L(Q,) are not normal for vocabularies of uncountable power 20,. It also follows that first-order logic is the only finite-dependence logic having normal model spaces and satisfying at the same time the downward Liiwenheim-Skolem theorem and uniform reduction for pairs.