Results on various closed sets in bi-generalised topological spaces

Abstract

The exploration and examination of bi-generalized topological spaces are driven by the aim of investigating wider generalizations of topological spaces, which facilitates a more adaptable framework for analyzing topological properties and their interrelations. The concept of bi- generalised topological space was introduced by Boonpok in 2010. The objective of this study is to provide some new results in the sets, namely (m, n)-closed sets and (m, n)-generalised closed sets in bi-generalised topological spaces. In addition, two new types of closed sets are defined: (m, n)-α＊-closed sets and (m, n) − β＊-closed sets, and some results for these new closed sets are provided. For a non-empty set X, the triple (X, μ₁, μ₂) is a bi-generalised topological space, where μ₁ and μ₂ are generalised topologies on X. The members of μ(m) are called μm-open sets, m =1, 2. The complement of μ(m)-open set is μm-closed set. A subset A of a bi-generalised topological space (X, μ₁, μ₂) is called (m, n)-closed if cl(μm)(cl(μn)(A)) = A , where m, n = 1,2 with m ≠ n . Similarly, the set A in X is called (m, n)-generalised closed if cl(μn) (A) ⊆ U, whenever A ⊆ U and U is μm-open set in X. In this study, firstly, it is shown that if the intersection or union of two subsets of a bi-generalised topological space is (m, n)-closed, then the subsets need not be (m, n)-closed. Next, a similar result for (m, n)-generalised closed sets are shown. Secondly, (m, n)α＊-closed set is defined as follows: In a bi-generalised topological space (X, μ₁, μ₂), a subset A in X is called (m, n)α＊-closed if int(μm)(cl(μn)(A)) ⊆U for every A ⊆ U and U is an (m, n)-open set. Then, it is shown that the intersection of two(m, n)-α＊closed sets need not be (m, n)-α＊closed. Also, if the intersection of two sets A and B is (m, n)-α＊closed, then the sets A, B need not be (m, n)-α＊closed. Similarly, it can be shown that the union of two (m, n)-α＊closed sets need not be (m, n)-α＊closed. Also, it is proved that a subset A in a bi-generalised topological space (X, μ₁, μ₂) is (m, n)-α＊closed if and only if X\A is (m, n)-α＊open. Finally, another closed set is introduced, namely (m, n)-β＊closed as follows: A set A in a bi-generalised topological space is called (m, n)-β＊closed if clμn(intμm(A)) ⊆ U for every A contained in U , where U is (m, n)-open set. Also, this study found that the intersection or union of two (m, n)-β＊closed sets need not be (m, n)-β＊closed.