WEAK SEPARATION AXIOMS VIA 𝑫𝝎, 𝑫𝜶−𝝎, 𝑫𝒑𝒓𝒆−𝝎, 𝑫𝒃−𝝎, AND 𝑫𝜷−𝝎 -SETS

Abstract

In this paper we define new types of sets we call them 𝐷𝜔, 𝐷𝛼−𝜔, 𝐷𝑝𝑟𝑒−𝜔, 𝐷𝑝𝑟𝑒−𝜔, 𝐷𝑏−𝜔, and 𝐷𝛽−𝜔 −sets and use them to define some associative separation axioms. Some theorems about the relation between them and the weak separation axioms introduced by M. H. Hadi in [1] are proved, with some other simple theorems. Throughout this paper , (𝑋, 𝑇) stands for topological space. Let (𝑋, 𝑇) be a topological space and 𝐴 a subset of 𝑋. A point 𝑥 in 𝑋 is called condensation point of 𝐴 if for each 𝑈 in 𝑇 with 𝑥 in 𝑈, the set U ∩ 𝐴 is uncountable [10]. In 1982 the 𝜔 −closed set was first introduced by H. Z. Hdeib in [10], and he defined it as: 𝐴 is 𝝎 −closed if it contains all its condensation points and the 𝝎 −open set is the complement of the 𝜔 −closed set. Equivalently. A sub set 𝑊 of a space (𝑋, 𝑇), is ω −open if and only if for each 𝑥 ∈ 𝑊 , there exists 𝑈 ∈ 𝑇 such that 𝑥 ∈ 𝑈and 𝑈\𝑊 is countable. The collection of all 𝜔 −open sets of (𝑋, 𝑇) denoted 𝑇𝜔 form topology on 𝑋 and it is finer than 𝑇. Several characterizations of 𝜔 −closed sets were provided in [3, 4, 5, 6].