Relatively Continuous Mappings of Topological Spaces
by
Mark Burgin
Dept. of Mathematics, UCLA

To achieve more adequate description of different phenomena, a new approach has been suggested. It is called the neoclassical analysis [M.Burgin, Neoclassical Analysis, Fuzzy Sets and Systems, 1995, v.75, No.2]. In it ordinary structures of analysis are studied by means of fuzzy concepts. For example, the continuous functions become a part of the set of the fuzzy continuous functions. It extends the scope of analysis making its methods more precise. Consequently, new results are obtained extending and completing classical theorems. In addition, analytical methods used for applications also become more powerful and efficient. The neoclassical analysis is developed for metric spaces. As there are important non-metrizable spaces, we have a problem how it is possible to define fuzzy continuity for a general topological spaces. Here, a solution of this problem is given. It is based on the concept of a discontinuity structure Q of a topological space X, which is a mapping of X corresponding to each point some filter of its neighborhoods. It makes it possible to define (Q, R)-continuous and R-continuous mappings from a space X with a discontinuity structure Q into a space Y with a discontinuity structure R. Fuzzy continuous functions are examples of R-continuous mappings. Different properties of (Q, R)-continuous mappings are obtained.

Date received: June 7, 1999