This paper presents a general unified approach to the notions of generalized closedness in topological spaces. The research concerning the notion of generalized closed sets in topological spaces was initiated by Norman Levine in 1970. In the succeeding years, the concepts of this type of generalizations have been investigated in many versions using the standard generalizations of topologies which has resulted in a large body of literature. However, the methods and results in the past years have become standard and lacking in innovation. The basic notion used in this conception is the closure operator designated by a family B⊆ P (X), which need not be a Kuratowski operator. Here, we introduce a general conception of natural extensions of families B⊆ P (X), denoted by B⊲ K, which are determined by other families K⊆ P (X). Precisely,