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 subfamil y of I(X) into I(X), subject to certain conditions. We identify all th e extensions of a fuzzy core to fuzzy Cech closure operators and to fu zzy Kuratowski operators. The main purpose of this theory is to charac terize, as we do here, some properties of functions as being equivalen t to their continuity with respect to certain fuzzy topologies on thei r domains and codomains. We nominate three existing theories as specia l instances of core theory. One is from General Topology, and is stron gly related to applications: the theory of the Scott topology on a par tially ordered set. The other two theories are from Fuzzy Topology: Kl ein's theory of alpha-closure operators, and the Mashhour et al. theor y of pseudo-closure operators.