It seems to me that the essential difference between graphs (including hypergraphs) and topological spaces resides in the difference between the covariant and contravariant power set functors. In both cases an object is a family of subsets of a set, with such subsets traditionally called respectively edges and open sets. But whereas a morphism of hypergraphs is a function for which the image of an edge is an edge, a continuous function is one for which the *inverse* image of an open set is open. Is this contrast discussed somewhere? And just how different does this make the resulting two concepts? (Treat the latter as being about *generalized* topological spaces, which drop the requirement that arbitrary unions and finite intersections of open sets be open.) Hypergraphs are a staple of the combinatorial literature, which seems to prefer small edges, doubletons being the motivating case definitive of the so-called simple graphs (of the undirected kind). Edges *coexist* in the sense that the whole hypergraph is viewed as the result of pasting its edges together. In contrast a topological space is perceived as more geometric than combinatorial. And open sets are not so much coexisting constituents of the space as *alternative* ways of "smoothly" partitioning the space into a closed and an open subset. Open sets are typically larger than edges, witness Euclidean space whose nonempty open subsets are all infinite. Whereas the natural view of an edge is an atomic constituent of the whole hypergraph, the natural view of an open set is as a (smoothly bounded) subspace. This passage from conjunctions of edges to disjunctions of open sets would appear to be a natural correlate of the passage from the covariant to the contravariant power set functor. My interest in this distinction stems from Chu's completion of a closed category to a self-dual closed (= *-autonomous) category with a specified dualizer. The morphisms of Chu's completion are defined contravariantly, so that when applied to Set it yields exactly these generalized topological spaces rather than hypergraphs. -- Vaughan Pratt +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++