A Unified Treatment of Generalized Closed Sets in Topological Spaces

Abstract

Abstract 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 ᐊ 𝒦, which are determined by other families 𝒦 ⊆ 𝒫(X). Precisely, ℬ ⊲ 𝒦 = { A ⊆ X : A ¯ ℬ ⊆ A ¯ 𝒦 } , \mathcal{B} \triangleleft \mathcal{K} = \left\{ {A \subseteq X:{{\bar A}^\mathcal{B}} \subseteq {{\bar A}^\mathcal{K}}} \right\}, where ( … ) ¯ 𝒜 {\overline {\left( \ldots \right)} ^\mathcal{A}} denotes the closure operator designated by 𝒜 ⊆ 𝒫(X). We prove that the collection of all generalizations ℬ ᐊ 𝒦, where ℬ, 𝒦 ⊆ 𝒫 (X), forms a Boolean algebra. In this theory, the family of all generalized closed sets in a topological space X(𝒯 )is equal to 𝒞 ᐊ 𝒯, where 𝒞 is the family of all closed subsets of X. This concept gives tools that enable the systemizing and developing of the current research area of this topic. The results obtained in this general conception easily extend and imply well-known theorems as obvious corollaries. Moreover, they also give many new results concerning relationships between various types of generalized closedness studied so far in a topological space. In particular, we prove and demonstrate in a graph that in a topological space X(𝒯) there exist only nine different generalizations determined by the standard generalizations of topologies. The tools introduced in this paper enabled us to show that many generalizations studied in the literature are improper.