757 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 M. A. Abdel Karim A. I. Nasir Abstract The notions Ǐ-semi-g-closedness and Ǐ-semi-g-openness were used to generalize and introduced new classes of separation axioms in ideal spaces. Many relations among several sorts of these classes are summarized, also. Keyword: Ideal, separation axioms, Ǐ , Ǐ , Ǐ , Ǐ . 1. Introduction Abdel Karim and Nasir [1]. Introduced the notion Ǐ set in ideal spaces. A set of ( Ǐ) is Ǐ set (artlessly,Ǐ ), when prerequisite Ǐ and is semi-open, leads to ( ) Ǐ. So, the set is nameable Ǐ (artlessly,Ǐ ) whenever is Ǐ set. The collection of all Ǐ (respectively, Ǐ ) sets in ( Ǐ) notate as Ǐ ( ) (respectively, Ǐ ( )) . Every open set in ( ) is Ǐ in ( Ǐ)[1]. And the reverse of this statement is incorrect [1]. The idea of ideals on a nonempty set was started by Kuratowski in 1933[2]. The ideal Ǐ on where Ǐ ( ) that is valid the preconditions finite additivity ( Ǐ Ǐ) and heredity ( Ǐ Ǐ) [2]. In 1945, Vaidynathaswamy [3]. Was the first mathematician who uses the idea of ideal in science of topology, which used to create a new generalization of topological spaces, which is called ideal topological space and notate as ( Ǐ)[4]. Since then, more authors are interested in this scope of the study. Many results have been achieved [5-8]. The concept of " - " was firstly offered by N.Levine [9]. A set in ( ) is called - if ( ( )) [9]. Recently, the main goal of our work is to construct generalizations of the notion separation axioms in topological space by using the notion of Ǐ set. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.1.2382 Department of mathematics, College of Education for Pure Sciences, Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq. Maherjaleel89@yahoo.com Ahmed_math06@yahoo.com Separation Axioms via Ǐ ­ Semi­ g­ Open Sets Article history: Received 2 June 2019, Accepted 4 July 2019, Publish January 2020. file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/Maherjaleel89@yahoo.com file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/Ahmed_math06@yahoo.com file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/Ahmed_math06@yahoo.com 758 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 2. Separation Axioms via Ǐsg-open Sets. Here, we will present new type of separation axioms in ideal spaces via Ǐ set. Also, relations between these sorts are discussed. The following definition is for the research who did this research. Definition 2.1. The triple ( Ǐ) is called Ǐ (shortly,Ǐ ), if for any element , there is an Ǐ set containing only one of them. Remark 2.2. If ( ) is then ( Ǐ) Ǐ . Remark 2.3. The below phrases are rewards; i. is Ǐ . ii. For each element , there is an Ǐ set containing only one of them. Definition 2.4. ( Ǐ) is called Ǐ (artlessly, Ǐ ), if for any elements , there are Ǐ sets, whenever ( ) , ( ). Remark 2.5. If ( ) is then ( Ǐ) Ǐ . Remark 2.6. ( Ǐ) is Ǐ whenever it is an Ǐ . The conclusions in Remark 2.6, is not reversible by example below. Example 2.7. The Ǐ ( Ǐ) (where * +, * * + * + * ++ and Ǐ * }), is not Ǐ . Remark 2.8. The below phrases are rewards; i. is Ǐ . ii. For each elements , there are two Ǐ sets and satisfy ( ) and ( ). Remark 2.9. For ( Ǐ), if * + is Ǐ set , then is Ǐ . The conclusions in Remark 2.9, is not reversible by example below. Example 2.10. Given ( Ǐ); ( * +, * + and Ǐ * * +}) is Ǐ , while * + is not Ǐ . Definition 2.11. ( Ǐ) is called Ǐ (artlessly, Ǐ ), if for any elements , there are Ǐ sets and satisfy , and . Remark 2.12. If ( ) is , then ( Ǐ) is Ǐ . Remark 2.13. A space ( Ǐ) is Ǐ whenever it is an Ǐ . The conclusions in Remark 2.13, is not reversible by example below. Example 2.14. A space ( ) when the set of all natural numbers, cof is the cofinite topology on and Ǐ * +. Clearly, that ( Ǐ) is Ǐ which is not Ǐ . We have previously noted that is Ǐ whenever it is ( ). The opposite is not generally achieved by example below. Example 2.15. ( Ǐ) is Ǐ ( ) where * +, * + and Ǐ is the class of all sets in . But the space ( ) is not . The following chart shows the relationships among the various types of notions of our previously mentioned. 759 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 3. Separation Axioms via Some Types of Functions In this portion we will review some sorts of functions which presented by Abdel Karim and Nasir [1]. And study their impact on the notions of separated axioms. Definition 3.1:[1]. ( Ǐ) ( ) is; i. , , if the precondition is Ǐ in , leads to ( ) is in . ii. , , if the precondition is open in , leads to ( ) is in iii. , , if the precondition is Ǐ in , leads to ( ) is open in . Proposition 3.2: is Ǐ ( ) and is a surjective from ( Ǐ) ( ) implies is . Proof: Follows from ( ) is in for all Ǐ set in . Proposition 3.3: is ( ) and is a surjective from ( Ǐ) ( ) implies is . Proof: Follows from ( ) is in for all set in . Proposition 3.4: is Ǐ ( ) and is a surjective from ( Ǐ) ( ) implies is . Proof: Follows from ( ) is in for all Ǐ set in . Remark 3.5: If is bijective open function from ( ) ( ) and is ( ), then ( ) is , for any ideal on . Definition 3.6:[1]. The function from ( Ǐ) ( ) is; i. , , if ( ) ( ) for every . ii. , notate as function, if ( ) for every ( ) . iii. , notate as , if ( ) ( ) for every ( ). Proposition 3.7: If is ( ) and ( Ǐ) ( ) is an injective then is Ǐ . Proof: Since ( ) ( ) for all . 𝑇 𝑠𝑝𝑎𝑐𝑒 𝑇 𝑠𝑝𝑎𝑐𝑒 𝑇 𝑠𝑝𝑎𝑐𝑒 Ǐ𝑠𝑔 𝑇 𝑠𝑝𝑎𝑐𝑒 Ǐ𝑠𝑔 𝑇 𝑠𝑝𝑎𝑐𝑒 Ǐ𝑠𝑔 𝑇 𝑠𝑝𝑎𝑐𝑒 761 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Corollary 3.8: If is ( ) and ( Ǐ) ( ) is an injective then is . Proof: Since, every continuous function is [1]. And then Proposition 3.7 is applicable. Proposition 3.9: If is ( ) and ( Ǐ) ( ) is an injective then is . Proof: Follows from, ( ) for all ( ). Proposition 3.10: If is ( ) and ( Ǐ) ( ) is an injective then is . Proof: Since ( ) ( ) for all ( ). 4. I- Semi- g- convergence sequence Sequences are one of important topics in all branches of mathematics, especially mathematical analysis and topology. So the convergence is the important property for the sequence [10, 11]. In this paragraph, we will use the notion of set to create new class of convergence. Definition 4.1. Given a space ( Ǐ) where and the sequence ( ) in . Then ( ) is called Ǐ to (artlessly, ) if for every Ǐsg-open set contained , where . A sequence ( ) is nameable Ǐ if it is not Ǐ . Proposition 4.2. If ( Ǐ) is Ǐ then every Ǐ sequence in has only one limit point. Proof: If we consider ( ) is a seq. in and so ; where . Since is Ǐ , then there are disjoint Ǐ -open sets such that and . Now, since and implies ; So, and implies ; Leads to where that contradiction. The precondition that a space is Ǐ is very requisite to make Proposition 4.2, is valid. Example 4.3. Consider ( Ǐ) where * +, * + and Ǐ * +. Obviously; the sequence ( ) in , where for all , has more than one limit point; that , and . Proposition 4.4. If a sequence ( ) is Ǐ to in an ideal space , then it is convergent to . Proof. Since every open set in ( Ǐ)is Ǐsg-open, then the proof is over. The meaningfulness in proposition 4.4, is not reversible, in general. Example 4.5. For an ideal space ( Ǐ), where , * + and Ǐ ( ). The sequence( ) , where , is convergent to which is notǏ . Proposition 4.6. Let be function from and ( ) be a sequence in , ( ) ( ) in whenever in . 767 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proposition 4.7. Let be function from and ( ) be a sequence in . ( ) ( ) in whenever in . Proposition 4.8. Let be function from and ( ) be a sequence in , ( ) ( ) in whenever in . 5. Conclusion The separation axioms appeared in new types and generalizations by used Ǐ-semi-g-openness, in addition a new types of convergence in ideal spaces were studied such as Ǐ , and the relationships between them was clarified by using several example. References 1. Abdel Karim, M.A.; Nasir, A.I. on I-semi-g-closed sets and I-semi-g-continuous functions. Journal of the Indian Mathematical society.2019. 2. Kuratowski, K. Topology. New York: Academic Press.1933, I. 3. Vaidyanathaswamy, V. The localization theory in set topology. Proc. Indian Acad. Sci. 1945, 20, 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, 1-10. 5. ALhawez, Z.T. On generalized -Closed set In Topological Spaces. Ibn Al-Haithatham Journal for Pure and Applied Science.2015, 28, 204-213. 6. Mahmood, S.I. on generalized Regular Continuous Functions in Topological Spaces. 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