This study proposes novel decomposed Fermatean fuzzy sets that incorporate membership and non-membership functions with functional and dysfunctional perspectives. First, definitions, basic arithmetic operations of decomposed Fermatean fuzzy sets are presented. Then, a novel decomposed Fermatean fuzzy TOPSIS method is developed to demonstrate the ability of the proposed decomposed Fermatean fuzzy sets to be modeled by multi-criteria decision making methods. In order to show its usefulness, an illustrative example is finally provided.