في هذا البحث قدمنا مفهوم دالة الهوية من النوع &nbsp;&nbsp;&nbsp;والتي هي الدالة &nbsp;من الفضاء التبولوجي&nbsp; الى فضاء تبولوجي اخر &nbsp;وكل من &nbsp;مؤثر على &nbsp;على التوالي فنقول ان الدالة &nbsp;هي دالة هوية من النوع &nbsp;اذا وفقط اذا كانت شاملة والصورة العكسية للمجوعة &nbsp;تكون -open في &nbsp;اذا وفقط اذا كانت المجموعة &nbsp;هي -open في .In this paper, we study and introduce the notion of -identification function which is a function &nbsp;from a topological&nbsp; space &nbsp;to another topological space &nbsp;and &nbsp;&nbsp;be two operators associative with&nbsp; &nbsp;&nbsp;respectively, then &nbsp;is called -identification function if and only if &nbsp;is onto and &nbsp;is -open set in &nbsp;if and only if &nbsp;is -open set in 

Al-Khafaji, Aqeel Ketab

English

Journals of University of Babylon

Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٣٠On (T,L)- Identification Function   Aqeel  Ketab  Al-Khafaji Department of Mathematics, Collage of education For Pure Sciences University of Babylon aqeelketab@gmail.com   Abstract:   In this paper, we study and introduce the notion of -identification function which is a function  from a topological  space  to another topological space  and   be two operators associative with    respectively, then  is called -identification function if and only if  is onto and  is -open set in  if and only if  is -open set in . Keywords: Identification function, -identification function, Operator, -open set,(T,L)- continuous function, -contra continuous function.        :الخالصةالى فضاء    من الفضاء التبولوجي والتي هي الدالة   في هذا البحث قدمنا مفهوم دالة الهوية من النوع  اذا وفقط   هي دالة هوية من النوع على التوالي فنقول ان الدالة على   مؤثر وكل من تبولوجي اخر   . في open- هي  اذا وفقط اذا كانت المجموعة في  open- تكون اذا كانت شاملة والصورة العكسية للمجوعة الدالـة  ، الدالة المستمرة من النوع ، المجموعة المفتوحة من النوع ، من النوع  دالة الهوية :مفتاحيةالكلمات ال   .ضد المستمرة من النوع 1. Introduction In 1966 HS [ Hu,1966] introduced the concept of an identification function (Definition 3.1) and in 1997  Al-K utaibi [Alkutabibi,1997] introduced the notion of semi-identification and some other types of identification. In this paper we introduce the notion of -identification function using the concepts of an operator associated to a topology (Definition 2.1) and -open set (Definition 2.2) was introduced by Kasahara [ Kasahara ,1979]   and Ogata [ Ogata, 1991] respectively . 2. Fundamental Concepts In this section we recall the basic definitions needed in this work. 2.1. Definition  [ Kasahara, 1979] Let  be a topological space, let  be the power set of  Let , we say that  is an operator associated with the topology  on  if  for every open set in  . We denote by  as a topological space with an operator  associated with the topology  and we will call it operator topological space. 2.2. Example   [ Mustafa, 2014] Let  be a topological space, Let  be defined as follows, Let  where Let  Notice that if Let  is open in , then  So that  is an operator associated with the topology  on  and the triple  is an operator topological space. If  is the identity operator  then the triple  will reduces to  , so that the operator topological space is the ordinary topological space     Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٣١2.3. Definition   [ Ogata,1991] Let  be an operator topological space, A subset of  is said to be -open set if for each  there exists an open set  containing  such that  A subset  is said to be -closed set  if it its complement is -open set.  It is clear that every -open set is open, but the converse is not necessarily true as shown in the following example : 2.4.  Example   Let  be an operator topological space, where and  is a closure operator. Consider  , we claim that   is open but not -open. Now, let  also  which is open in  Let  then  that means . But , which means that  is not -open .  2.5. Definition     [ Ogata. 1991] Let and  be two operator topological spaces. We say that a function  is (T,L)- continuous function if for each point  and every open set  containing  ther exists an open set  containing  such that  . 2.6.  Theorem      [ Mustafa, 2014] Let  is - continuous function, then the inverse image of each -open set in  is -open set in . Proof:  Let  be -open set in , to prove that,  is -open set in Let  then,  but  is -open set then there exists an open set   in  such that  Now is -continuous at  then there exists an open set  in  containing  such that,  so,  Now, that is,  which means that  is -open.  2.7. Definition    [ Ogata, 1991] Let and  be two operator topological spaces. We say that a function  is -contra continuous function if  is -closed set in  for all -open set  in  3.The Main Results 3.1.  Definition   [ Hu, 1966]   Let  be two topological spaces. A function   is called identification function if and only if (i)  is onto and (ii) is open set in  if and only if  is an open set in  . 3.2. Example   Let and be tow topological spaces  where  and ,  , let  be define by  , then  is identification function .  3.3. Definition         A function  from the operator topological space to another topological space  is called  -identification function if and only if  is onto and  is -open set in  if and only if  is -open set in . Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٣٢ 3.4. Example       Let and  be two operator topological spaces, where  ,  such that                  and  , . Now the set of all  -open set  and the set of all  -open set  , let  be define by , then  is -identification function. 3.5. Remark    (i)  Observe that if   and  are the identity operators on  and  respectively, then the Definition 2.3  is reduced to the definition  identification function (Definition  2.1). (ii)  Identification function  and -identification function are independent. And the following two examples will show that.  3.6. Example       Let  ,  and  defined as     and   defined as   then the  set of all -open set and the set of all -open set . Let   is identity function, then  is identification function but not  -identification function [take  is -open set in , then  which is not -open set in  . 3.7. Example      Let and  be two operator topological spaces, where  ,  such that                  and  , such that  .  Now the set of all  -open set  and the set of all  -open set , let  is identity function, then  is -identification function but not identification function[take  which is not open set in   3.8. Theorem         An onto function  is -identification function if and only if is -closed set in  if and only if  is -closed set in  Proof:  Let be  -closed set in , then  is -open set in  but  is -identification function, then  is onto and  is -open set in , thus  is -closed in . Similarly, if  is -closed set in , then  is -open set in  and since  is  -identification function , then  is -open set in  and hence  is  -closed set in . Let be  -open set in , then  is -closed set in , then  is -closed set in  That is   is -open set in  Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٣٣Similarly, if   is -open set in  then  is -closed set in , and then  is -closed set in  and hence  is  -open set in  Since  is onto then  is  -identification function. 3.9. Definition   A function  from the operator topological space to another topological space  is called  - open (closed)  if and only if the image of every  -open ( -closed) set in  is -open ( - closed) set in . Example  3.10 Let and  be two operator topological spaces, where    and ,  , where  and  are defined as follows:   , then the set of all  -open set     and the set of all  -open set   let  be define by , then  is -open  function.  3.11. Theorem            If  is, - open, onto and - continuous function, then  is  -identification function. Proof: Let  be a subset of  such that  is -open set in . Since  is onto, we have  , since   is -open set in  and  is - open, then  is -open  set in  Now  is -open set in , since  is - continuous function, then  is -open set in . That is  is  -identification function. 3.12. Corollary       A function  is -identification function, if it is, - closed, onto and - continuous function. Proof:  Clear  3.13. Theorem    Let   and  are operator topological spaces, then if:  is -identification function, and   is -identification function,then  is -identification function . Proof:  Clear that, the composition of two onto function is onto. Now , let  be any -open set in , since  is -identification function, then  is -open set in , and  is - identification function, we have is -open set in   Similarly, if  is -open set in , since  is - identification function, then  is -open set in , and since  is -identification function, then  be any -open set in . Thus  is - identification function  3.14. Theorem          Let  is -identification function, and   is a function, then the following statements are valid: (i) If   is - continuous function, then - continuous function. Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٣٤(ii) If   is -contra continuous function, then  is -contra continuous function. Proof:      Assume that     (i) let  be -open set in , put  and , since , we have  Since  is  -open set in  and  is - continuous function, then  is -open set in , this means that  is -open set in . But  is -identification function, then  is -open set in , that is  is -open set in . Thus - continuous function  (ii) let  be -open set in , put  and , since , we have  Since  is -contra continuous function, then  is -closed set in , that is  is -closed set in . Since , then  is -closed set in . But  is -identification function, then   is -closed set in  (Theorem 3.8), that is   is -closed set in . Thus - contra continuous function  4.Future works: We will discuss the following concepts : 1-  –semi-identification function. (Using the concepts operator  and semi-open sets) 2- - pre-identification function. (Using the concepts operator  and preopen sets) 3- We can use the concept of -open set and the concept of operator  to define - -identification function. References  Alkutaibi, S. H., 1997, On Some Types Of Identifications, Journal of sciences, college of education, Tikrit University, Vol. 4 No.2. 157-163.  Al-jizani, A. K., 2004,  The Operator   And New Types Of Open Sets And Spaces, M.Sc. Thesis,  Mu'tah Uneversity, Jordan.   Carpintero, C, Rosas E and Vielma J., 2001, Generalization Functions Contera-Continuas, Divulgaciones Mathematics . Vol. 9, No. 2. 33-44.  Kasahara, S.,1979,Operation-Compact Spaces, Mathematical Japonica.Vol. 21.  97-105.  Mustafa, H. J., Al-Khafaji,A.K. and Al-Hindawe,A.L., Operator Topological Spaces, Journal of college of education, Mustansiriyah University, Vol. 1 NO.3,2014, 225-232.  Hu, S. T., 1966, Introduction to General Topology. Holden-day, Inc.   Ogata, H., 1991, Operation On Topological Spaces And Associated Topology, Math. Japonica, Vol. 36 No. 1.175-184.           

On (T,L)- Identification Function

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On (T,L)- Identification Function

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