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The primary objective of this research work is to generalize the study of e-chains between two points to two sets in metric spaces. This generalization yields a characterization of e-chainable sets in terms of e-chains between points and sequences, In chapter 3, we investigate the relation between compact sets and e- chainable sets and define self- chainable sets, uniform chainable sets, strongly chainable sets with examples and prove fixed point theorem. After this in chapter 4, we define locally chainable sets in metric space and prove various new results. In chapter 5, The notion of locally chainable metric spaces is introduced. In chaper 6, The concept of paths and homotopy in metric spaces have been generalized to c-path and c-homotopy and path chainable spaces have been defined. In the last chapter, we extend the concept of e-chains in the 2- metric space and prove their characterizations in terms of chainable sets.