On the core of a collection of coalitions

For a collection Omega of subsets of a finite set N we define its core to be equal to the polyhedral cone (x is an element of IRN : Sigma(i is an element of N) x(i) = 0 and Sigma(i is an element of s) x(i) greater than or equal to 0 for all S is an element of Omega). This note describes several applications of this concept in the field of cooperative game theory. Especially collections Omega are considered with core equal to {0}. This property of a one-point core is proved to be equivalent to the non-degeneracy and balancedness of Omega. Further, the notion of exact cover is discussed and used in a second characterization of collections Omega with core equal to {0}.