Continuous logic in a classical setting

Let L be a first-order two-sorted language and consider a class of L-structures of the form M, X where M varies among structures of the first sort, while X is fixed in the second sort, and it is assumed to be a compact Hausdorff space. When X is a compact subset of the real line, one way to treat classes of this kind modeltheoretically is via continuous-valued logic, as in Ben Yaacov et al. (Model theory for metric structures, Cambridge University Press, 2010). An earlier approach due to Henson and Iovino (Ultraproducts in analysis, analysis and logic (Mons, 1997), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2002) is based on the notion of positive formulas and is tailored to the model theory of Banach spaces. Here we show that a similar approach is possible for a more general class of models. We introduce suitable versions of elementarity, compactness, saturation, quantifier elimination and other basic tools.