From: "Allen Knutson" <aknaton@math.Princeton.EDU> Date: Thu, 28 Jan 93 19:31:28 EST > Whereas an edge is a subobject, an open set is a congruence class. Mm? I would've called a closed set a congruence class - putting on an equivalence relation should be like taking a (Hausdorff) quotient, which unifies a closed subset, not an open one. So I still think open sets are a red herring for your purposes. Not at all. The distinction between open and closed sets is "static" in the sense that it applies only to the objects of Top, not their morphisms. That the inverse image of a continuous function takes open sets to open implies that it takes closed sets to closed, *and conversely*. The generalization from topological spaces (T,O) to hyperspaces (T,O) (still a set of *points* paired with a set of its subsets called the *open sets*), obtained by dropping the requirement that unions and finite intersections of opens are open, removes even the static distinction. It is then quite arbitrary whether one calls the distinguished subsets of a hyperspace open or closed. I chose the former, but I'm open to arguments for calling them closed. This symmetry of open and closed does not obtain for hypergraphs (V,E), whose objects are as for hyperspaces (but now viewed as a set of *vertices* paired with a set of its subsets called the *edges*) but whose morphisms f:V->V' are such that the *direct* image of an edge in E is an edge in E'. This does not imply that the direct image of the complement of an edge is the complement of an edge. Whereas both inverse and direct images are complete semilattice maps, the inverse image preserves complements (i.e. is a CABA map) while the direct image instead preserves atoms. It is the preservation of complements that gives both open and closed sets the characteristics of a congruence class. The preservation of atoms makes edges look instead like subalgebras, but it does not make complements of edges look like subalgebras. This is a crucial difference between edges and open sets. -- Vaughan Pratt ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++