Fuzzy cores and their extensions (Q1313047)

A fuzzy core on a carrier set \(X\) is defined as an operator on a subfamily of \([0,1]^ X\) into \([0,1]^ X\) subject to conditions motivated by the definition of a Čech closure. The author notes that fuzzy Čech and Kuratowski operators induce fuzzy cores and examines several ways of extending a core operator to all fuzzy subsets of the carrier. These extensions lead to six notions of continuity for maps between fuzzy cores, and interconnections among them are established. Lastly, the author shows that the Scott topology, Klein's generating families of \(\alpha\)-closure operators, and Mashhour's pseudo-closure operators can be obtained from fuzzy cores.