and Kuratowski's axioms for c are theorems. Likewise, given an axiomatization of c that is analogous to C5-7, i may be defined in terms of c, and C5-7 become theorems. Adding C5-7 to GEMT results in Casati and Varzi's preferred mereotopological theory, GEMTC.x is self-connected if it satisfies the following predicate: Note that the primitive and defined predicates of MT alone suffice for this definition. The predicate SC enables formalizing the necessary condition given in Whitehead's Process and Reality for the mereological sum of two individuals to exist: they must be connected. Formally: C8. Given some mereotopology X, adding C8 to X results in what Casati and Varzi call the Whiteheadian extension of X, denoted WX. Hence the theory whose axioms are C1-8 is WGEMTC. The converse of C8 is a GEMTC theorem. Hence given the axioms of GEMTC, C is a defined predicate if O and SC are taken as primitive predicates. If the underlying mereology is atomless and weaker than GEM, the axiom that assures the absence of atoms (P9 in Casati and Varzi 1999) may be replaced by C9, which postulates that no individual has a topological boundary: C9. When the domain consists of geometric figures, the boundaries can be points, curves, and surfaces. What boundaries could mean, given other ontologies, is not an easy matter and is discussed in Casati and Varzi (1999: ch. 5). See also Mereology Pointless topology Point-set topology Topology Topological space (with links to T0 through T6) Whitehead's point-free geometry Notes References Biacino L., and Gerla G., 1991, "Connection Structures," Notre Dame Journal of Formal Logic 32: 242–47. Casati, Roberto, and Varzi, Achille, 1999. Parts and places: the structures of spatial representation. MIT Press. Stell J. G., 2000, "Boolean connection algebras: A new approach to the Region-Connection Calculus," Artificial Intelligence'' 122: 111–136. External links Stanford Encyclopedia of Philosophy: Boundary—by Achille Varzi. With many references. Mathematical axioms Mereology Ontology Topology