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 ð·ðœ”, ð·ð›¼âˆ’ðœ”, ð·ð‘ð‘Ÿð‘’−ðœ”, ð·ð‘ð‘Ÿð‘’−ðœ”, ð·ð‘−ðœ”, and ð·ð›½âˆ’𜔠−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 , (ð‘‹, ð‘‡) stands for topological space. Let (ð‘‹, ð‘‡) be a topological space and ð´ a subset of ð‘‹. A point ð‘¥ in ð‘‹ is called condensation point of ð´ if for each 𑈠in 𑇠with ð‘¥ in ð‘ˆ, the set U ∩ ð´ is uncountable [10]. In 1982 the 𜔠−closed set was first introduced by H. Z. Hdeib in [10], and he defined it as: ð´ is ðŽ −closed if it contains all its condensation points and the ðŽ −open set is the complement of the 𜔠−closed set. Equivalently. A sub set ð‘Š of a space (ð‘‹, ð‘‡), is ω −open if and only if for each 𑥠∈ ð‘Š , there exists 𑈠∈ 𑇠such that 𑥠∈ ð‘ˆand ð‘ˆ\ð‘Š is countable. The collection of all 𜔠−open sets of (ð‘‹, ð‘‡) denoted ð‘‡ðœ” form topology on ð‘‹ and it is finer than ð‘‡. Several characterizations of 𜔠−closed sets were provided in [3, 4, 5, 6].