251 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Ahmed. A. Jassam R. B. Esmaeel Abstract The main aim of this paper is to use the notion ـ ـ ـ which was introduced in [1]. To offered new classes of separation axioms in ideal spaces. So, we offered new type of notions of convergence in ideal spaces via the ـ ـ ـ set. Relations among several types of separation axioms that offered were explained. Keyword: Ideal, separation axioms, ـ ـ ـ , ـ ـ ـ , ـ ـ ـ , . ـ ـ ـ , ـ ـ ـ 1. Introduction In 1933, Kuratowski [2]. Presented the concept of ideals on non-empty sets. A collection is namely an ideals on a nonempty set when the following two conditions are met; (i) whenever and and (ii) whenever and are belong to . Vaidyanathaswamy [3]. Had offered for initial the idea of ideal spaces by introduced the set operator namely local function. So he founded new generalize of the topological spaces, namely ideal space and symbolizes by , [4, 5]. The concept of "preـopen set" was introduced by Mashhour, Abd El- Monsef and El- Deeb, a set in is a pre-open when [6]. From that time many researchers have submitted many studies in this field [7-9]. Latterly, Ahmed and Esmaeel [1]. had submitted the concept of ـ ـ ـ (simply, ـ A set in is So, the set . ـ and is pre-open set, implies to ـ if the condition ـ in is namely ـ ـ ـ (simply, ـ if ـ is ـ The collection of all ـ (respectively, ـ in simply ـ (respectively, ـ ). For a space , ـ is finer than [1]. The main target of this article is to introduce new kinds of separation axioms in ideal spaces by using the notion ـ ـ ـ ـ .2 This portion is to submit new classes of separation axioms by using the notion of ـ . Properties of these sorts are studied and the relations between it are discussed.also. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.1.2381 ahm7a7a@gmail.com ranamumosa@yahoo.com Article history: Received 1 June 2019, Accepted 10 July 2019, Publish January 2020. CONVERGENCES VIA ẛ ـPRE ـg ـOPEN SET Department of Mathematics, College of Education for Pure Science, Ibn-Al Haitham, University of Baghdad, Iraq file:///C:/Users/Mustafa/Desktop/2020/رياضيات/ahm7a7a@gmail.com file:///C:/Users/Mustafa/Desktop/2020/رياضيات/ahm7a7a@gmail.com file:///C:/Users/Mustafa/Desktop/2020/رياضيات/ahm7a7a@gmail.com file:///C:/Users/Mustafa/Desktop/2020/رياضيات/ahm7a7a@gmail.com file:///C:/Users/Mustafa/Desktop/2020/رياضيات/ranamumosa@yahoo.com file:///C:/Users/Mustafa/Desktop/2020/رياضيات/ranamumosa@yahoo.com 251 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Definition 2.1: A space is namely (frugally, ) if for each elements , there exist an set containing only one of them. When is , it will lead to that is for any ideal on . Remark 2.2: For a space , the below sentences are rewards; i. Is an . ii. For each element , there is an set containing only one of them. Definition 2.3: A space is namely (frugally, ) if for each elements , there are ẛ sets and , satisfies ـ and . ـ When is , it will lead to that is for any ideal on . Remark 2.4: If is , implies that . The inverse meaning implied in Remark 2.4, does not valid, in general. Example 2.5: A space is a where , { } and . The space is not , since for the elements , there is no ـ set containing which does not contain . Remark 2.6: For a space , the below sentences are rewards; i. Is an . ii. For each elements , there are two sets and , such that ـ and ـ . Remark 2.7: If {r} is set for each in Ҳ, then is . Definition 2.8: A space is namely (frugally, ), if for each elements , there are disjoint sets and satisfies and . Clearly; if is implie that is , for any ideal on . Remark 2.9: If the space is then it is . The inverse meaning implied in Remark 2.9, does not valid, in general. Example 2.10: The ; such that , and is not . Since, for the elements , there are no disjoints ـ sets and such that and . Remark 2.11: For a space , the below sentences are rewards; i. is an . ii. For each elements , there are disjoint sets and , satisfies and . We have the truth that confirms that if is a ( ), then the ideal space is a . But the inverse meaning implied may be invalid, as shown in the following diagram. Example Below shows the relationships between the unlike classes of notions that presented previously. Ҳ is Ҳ is e Ҳ is Ҳ is Ҳ is Ҳ is 251 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Example 2.12: The where , and is not (where and ). 3. Separation axioms by using some types of function In this part, we will using some types of functions that we were offered it in Ahmed and Esmaeel [1]. And study the notions of new separation axioms under influence of these functions. Definition 3.1: [1]. A function Ὑ is i. , symbolizes , if is set in Ὑ whenever is set in Ҳ. ii. , symbolizes , if is set in Ὑ whenever is open set in Ҳ. iii. , symbolizes , if is open in Ὑ whenever is set in Ҳ. Proposition 3.2: If is an (respectively, and ) and Ὑ is surjective, then Ὑ is (respectively, and ). Proof: Since is in Ὑ whenever is set in Ҳ. Proposition 3.3: If a space is (respectively, and ) and Ὑ is surjective, then Ὑ is (respectively, and ). Proof: Since is set in Ὑ whenever is open in Ҳ. Proposition 3.4: If a space is (respectively, and ) and Ὑ is surjective, then Ὑ is (respectively, and ). Proof: Since is open in Ὑ whenever is set in Ҳ. Remark 3.5: If Ὑ is a bijective open function and a space is a (respectively, and ), then the space Ὑ is a (respectively, and ), for any ideal on Ὑ. Definition 3.6: [2]. A function Ὑ is; i. ẛ , symbolizes , if for all . ii. ẛ Symbolizes ẛ if , for all Ὑ . iii. ẛ , symbolizes ẛ , if for all Ὑ . Proposition 3.7: If Ὑ is (respectively, and ) and Ὑ is injective, ـ function, then is an (respectively, and ). Proof: Since for all . Corollary 3.8: If a space Ὑ is (respectively, and ) and Ὑ is injective, continuous function then is an ـ (respectively, and ). Proof: Clearly, the continuity leads to ـ continuity [2]. So Proposition 3.7 is valid. 255 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proposition 3.9: If is (respectively, and ) and Ὑ is injective, ـ then the space is (respectively, and ). Proof: follows by the result , for all Ὑ . Proposition 3.10: If a space Ὑ is a (respectively, and ) and Ὑ is an injective, function, then is an (respectively, and ). Proof: follows by the result if for all Ὑ ـ .4 In this part we will use the notion ـ to erection some class of convergence in ideal spaces namely ـ . So, the action of some sorts of functions are discussed like, ـ , ẛ and ẛ [1]. Definition 4.1: Let be an ideal space, and be a sequence in . Then is namely ـ to (frugally, ) if for every ـ set contained , such that A sequence is namely ـ if it is not ـ . Proposition 4.2: If is ـ then every ـ sequence in has a unique limit point. Proof: Let be a sequence in where and ; where . Since is ـ ـ , then such that and , where . Since and leads to ; So and leads to ; . Hence, , that is contradiction. The precondition that a space is ـ ـ is very requisite to make Proposition 4.2, is valid. Example 4.3: For a space where , and . Obviously; the sequence in , where for all , has three limit points; that , and . In mathematics, convergence sequence was an important subject [10, 11]. The following proposition explains the relationships between convergence and ـ to . Proposition 4.4: If a sequence is ـ to in , then it is convergence to . Proof: Since every open set in is ẛpg-open, then the proof is over. The meaningfulness in Proposition 4.4, cannot be inverting, in general. Example 4.5: For a space , where set of all neutral numbers, and . The sequence , where , is convergence to which is not ـ . Proposition 4.6: Let Ὑ be ـ function and be a sequence in . If in , then in Proposition 4.7: Let Ὑ be ـ function and be a sequence in . If in , then in 251 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proposition 4.8: Let Ὑ be ـ ـ function and be a sequence in . If in , then in 5. Conclusion The notion ـ ـ ـ was use to offered new classes of separation axioms and new type of convergence in ideal spaces. Some relations and examples among several types of separation axioms that offered were explained. References 1. Ahmed, A.; Esmaeel R.B. on ẛ-pre-g-closed sets and ẛ-pre-g-continuous functions. Ready to be published in the Journal of the Indian Mathamatical society, 2019. 2. Kuratowski, K. Topology. New York: Acadeic Press.1933, I. 3. Vaidyanathaswamy V.; the Localization theory in set topology, proc. Indian Acad. Sci. 1945, 20, 1, 51-61. 4. Abd El- Monsef, M.E.; Nasef, A.A.; Radwan, A.E.; Esmaeel, R.B. On α- open sets with respect to an ideal. Journal of Advances studies in Topology.2014, 5, 3, 1-9. 5. Nasef, A.A.; Radwan, A.E.; Esmaeeln, R.B. Some properties of α-open sets with repect to an ideal, Iternational Journal of Pur and Applied Mathematics.2015, 102, 4, 613-630. 6. Mashhour, A.S.; Abd El- Monsef, M.E.; El- Deeb, S.N. on pre topological Spaces. Bull. Math. Dela Soc. R.S. de roumanie.1984, 28, 3945 ـ. 7. Dawood, N.A.; Nasir, A.I. Some Types of compactness in Bitopological Spaces and it is Associated Topology of α-subsets. Indian J. pure Appl. Math.2010, 27, 995-1004. 8. Mahmood, S.I. On generalized Regular Continuous Functions in Topological Spaces. Ibn Al-Haithatham Journal for Pure and Applied Science.2017, 25, 3, 377-376. 9. ALhawez Zinah, .T. On generalized -Closed set In Topological Spaces. Ibn Al- Haithatham Journal for Pure and Applied Science.2015, 28, 204-213. 10. Dugundji, J. Topology. University of South California, Los Angeles, Allyn and Bacon, Inc., Boston, Mass, 1966. 11. Munkres, J.R. Topology. Prentice-Hall, Inc., Englwood Cliffs, NewJersey, 1975.