IJPAM: Volume 16, No. 4 (2004)

DISCONTINUITY STRUCTURES
IN TOPOLOGICAL SPACES

M. Burgin
Department of Mathematics
University of California
405 Hilgard Avenue
Los Angeles, CA 90094-1555, USA


Abstract.In this paper, topological spaces are enriched by additional structures in order to give a more realistic representation of real and computational phenomena and at the same time, to provide for utilization of the powerful technique developed in topology. The suggested approach is based on the concept of a discontinuity structure Q of a topological space $ X $. This structure serves as an approximation to the initial topology on $ X $. Problems of science and engineering need such an approximation because all measurements and the majority of computations are performed approximately. Taking a mapping of a topological space $ X $ with the discontinuity structure Q into a topological space $ Y $ with a discontinuity structure R, we define (Q, R)-continuity and R-continuity of this mapping. Fuzzy continuous functions, which are studied in neoclassical analysis, are examples of R-continuous mappings. Different properties of (Q, R)-continuous mappings are obtained in the first part of this paper. In the second part, Q-connectedness is introduced and studied. This concept extends the conventional notion of connected sets in topological spaces and enables consider sets that are connected only to some extent, that is, in particular that the gaps between the components of such sets are not too big or we do not have precise knowledge about these gaps. In addition, a related concept of path Q-connectedness is introduced and studied.

Received: August 1, 2004

AMS Subject Classification: 54A05

Key Words and Phrases: topology, relative continuity, discontinuous structure, relative connectedness, fuzzy limit

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 16
Issue: 4