WEAK SEPARATION AXIOMS VIA ð‘«ðŽ, ð‘«ðœ¶âˆ’ðŽ, ð‘«ð’‘ð’“ð’†âˆ’ðŽ, ð‘«ð’ƒâˆ’ðŽ, AND ð‘«ðœ·âˆ’ðŽ -SETS - Hadi, Mustafa Hasan (European Scientific Journal, ESJ, 2013)
In this paper we define new types of sets we call them ð·ðœ”, ð·ð›¼âˆ’ðœ”, ð·ð‘ð‘Ÿð‘’âˆ’ðœ”, ð·ð‘ð‘Ÿð‘’âˆ’ðœ”, ð·ð‘âˆ’ðœ”, and ð·ð›½âˆ’ðœ” âˆ’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 ð‘ˆ in ð‘‡ with ð‘¥ in ð‘ˆ, the set U âˆ© ð´ is uncountable [10]. In 1982 the ðœ” âˆ’closed set was first introduced by H. Z. Hdeib in [10], and he defined it as: ð´ is ðŽ âˆ’closed if it contains all its condensation points and the ðŽ âˆ’open set is the complement of the ðœ” âˆ’closed set. Equivalently. A sub set ð‘Š of a space (ð‘‹, ð‘‡), is Ï‰ âˆ’open if and only if for each ð‘¥ âˆˆ ð‘Š , there exists ð‘ˆ âˆˆ ð‘‡ such that ð‘¥ âˆˆ ð‘ˆand ð‘ˆ\ð‘Š is countable. The collection of all ðœ” âˆ’open sets of (ð‘‹, ð‘‡) denoted ð‘‡ðœ” form topology on ð‘‹ and it is finer than ð‘‡. Several characterizations of ðœ” âˆ’closed sets were provided in [3, 4, 5, 6].