FUZZY CORES AND THEIR EXTENSIONS

We introduce Core Theory, simultaneously for General Topology and for Fuzzy Topology. A fuzzy core on a set X is an operator from a subfamily of I(X) into I(X), subject to certain conditions. We identify all the extensions of a fuzzy core to fuzzy Cech closure operators and to fuzzy Kuratowski operators. The main purpose of this theory is to characterize, as we do here, some properties of functions as being equivalent to their continuity with respect to certain fuzzy topologies on their domains and codomains. We nominate three existing theories as special instances of core theory. One is from General Topology, and is strongly related to applications: the theory of the Scott topology on a partially ordered set. The other two theories are from Fuzzy Topology: Klein's theory of alpha-closure operators, and the Mashhour et al. theory of pseudo-closure operators.