A derived set is a concept in topology that refers to the set of all limit points of a given set. A limit point of a set is a point that can be approached by other points from that set, meaning that every neighborhood around the limit point contains at least one point from the set, distinct from the limit point itself. For example, if we have a set A in a topological space, the derived set A' includes all the limit points of A. Derived sets help in understanding the closure and boundary properties of sets in topology, which are essential for studying continuity and convergence in mathematical analysis.