CATEGORICITY AND TOPOLOGICAL GRAPHS

Abstract. Let X be a topological graph; i.e., a union of finitely many points and arcs, with arcs joined only at end points. If Y is any locally connected metrizable compactum that is co-elementarily equivalent to X, then Y is homeomorphic to X. In particular, X and Y are homeomorphic if some lattice base for one is elementarily equivalent to some lattice base for the other. 1. introduction This paper is about the model-theoretic topology of compact Hausdorff spaces— also referred to as compacta—and our aim is to show that any topological graph is categorical, relative to the class of compacta that are locally connected and metrizable. As the term categorical has a range of interpretations, we begin with a general description of how it is used here. Suppose K is a class of objects, together with two reflexive symmetric relations, one finer than the other. To keep things straight, call the finer relation indistinguishability (always an equivalence relation) and the coarser one similarity (usually an equivalence relation). An object X ∈ K is defined to be categorical, relative to K, if any member of K that is similar to X is actually indistinguishable from X. 2000 Mathematics Subject Classification. 03C20,03C35,54F25,54F50. Key words and phrases. Categoricity, compacta, continua, locally connected, topologica