Metric structures are like first-order structures except that the formulas take truth values in the unit interval, and instead of equality there is a distance predicate with respect to which every function and predicate is uniformly continuous. Pre-metric structures are similar the distance predicate is only a pseudo-metric. In recent years the model theory of metric and pre-metric structures has been successfully developed in a way that is closely parallel to first order model theory, with many applications to analysis.
We consider general structures, where formulas still have truth values in the unit interval, but the predicates and functions need not be continuous with respect to a distance predicate. It is shown that every general structure can be expanded to a pre-metric structure by adding a distance predicate that is a uniform limit of formulas. Moreover, any two such expansions have the same notion of uniform convergence. This can be used to extend almost all of the model theory of metric structures to general structures in a precise way. For instance, the notion of a stable theory extends in a natural way to general structures, and the main results carry over.