For all the success of first order model theory, there are a host of mathematical objects which, defying first order axiomatization, lie outside of its scope: Banach spaces and Artinian rings, for example. The abstract elementary class (AEC) provides a unifying context in which to analyze the model theory of such classes of objects and, more generally, of non-first order logics. Obtained by taking the notion of elementary class, abandoning syntax, and retaining only the purely diagrammatic, category-theoretic properties of elementary embedding, they require a new notion of type--Galois type--which must necessarily be characterized in non-syntactic terms. Given this change, and given that we cannot fall back upon compactness of the underlying logic, classical methods and results rarely generalize to this framework. We present a near-exception to this: we exhibit a way of topologizing sets of Galois types (analogous to the syntactic topologies familiar from classical model theory), thereby providing a foothold for the definition of Cantor-Bendixson rank for Galois types and, ultimately, a family of Morley-like ranks. As an application, we use the latter to prove a generalization of a Galois stability result due to Baldwin, Kueker and VanDieren.