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This research paper is intended as a survey article on new types of mixing, weakly mixing, transitive and chaotic sets in a topological dynamical system. I mention here some new kinds of chaotic maps and chaotic sets and indicate their connection between them. Chaotic behavior is a manifestation of the complexity of nonlinear system. There are several different definitions of topological transitive sets and chaos, which describe the complexity of systems in different aspects. The present work mainly deals with some new types of these definitions. In this project, I explain the main new ingredients of new type of chaotic sets and transitive sets in a given topological space. Further I illustrate relationship between the new and basic definitions of sets are given.

New Types of Lambda-Transitive and Weakly Lambda-Mixing Sets