A Relation-Algebraic Approach to L - Fuzzy Topology
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Any science deals with the study of certain models of the real world. However, a model is always an abstraction resulting in some uncertainty, which must be considered. The theory of fuzzy sets is one way of formalizing one of the types of uncertainty that occurs when modeling real objects. Fuzzy sets have been applied in various real-world problems such as control system engineering, image processing, and weather forecasting systems. This research focuses on applying the categorical framework of abstract L - fuzzy relations to L-fuzzy topology with ideas, concepts and methods of the theory of L-fuzzy sets. Since L-fuzzy sets were introduced to deal with the problem of approximate reasoning, t − norm based operations are essential in the definition of L - fuzzy topologies. We use the abstract theory of arrow categories with additional t − norm based connectives to define L - fuzzy topologies abstractly. In particular, this thesis will provide an abstract relational definition of an L - fuzzy topology, consider bases of topological spaces, continuous maps, and the first two separation axioms T0 and T1. The resulting theory of L - fuzzy topological spaces provides the foundation for applications and algorithms in areas such as digital topology, i.e., analyzing images using topological features.