New Operators in Ideal Topological Spaces and Their Closure Spaces

Aksaray University Journal of Science and Engineering e-ISSN: 2587-1277 http://dergipark.gov.tr/asujse http://asujse.aksaray.edu.tr Aksaray J. Sci. Eng. Volume 3,Issue 2, pp. 112-128 doi: 10.29002/asujse.605003 Available online at Research Article New Operators in Ideal Topological Spaces and Their Closure Spaces Shyamapada Modak*, Md. Monirul Islam Department of Mathematics, University of Gour Banga, Malda 732 103, West Bengal, India ▪Received Date: Aug 10, 2019 ▪Revised Date:Nov 8, 2019 ▪Accepted Date:Dec 19, 2019 ▪Published Online: Dec 23, 2019 Abstract  In this paper, we introduce two operators associated with  and * * operators in ideal topological spaces and discuss the properties of these operators. We give further characterizations of Hayashi-Samuel spaces with the help of these two operators. We also give a brief discussion on homeomorphism of generalized closure spaces which were induced by these two operators. Keywords Ideal topological spaces,  -operator,  -operator, Hayashi-Samuel space, isotonic spaces, homeomorphism. 1. INTRODUCTION The study of local function on ideal topological space was introduced by Kuratowski [1] and Vaidyanathswamy [2]. The mathematicians like Jankovic and Hamlett [3, 4], Samuel [5], Hayashi [6], Hashimoto [7], Newcomb [8], Modak [9, 10], Bandyopadhyay and Modak [11, 12], Noiri and Modak [13], Al-Omari et al. [14, 15, 16, 17] have enriched this study. Natkaniec in [18] have introduced the complement of local function and it is called  -Operator. In an ideal topological space ( X , , ) , the local function ()* is defined as: A* ( , ) (or, simply, A* ) = {x  X : U x  A  } , where U x  ( x) , the collection of all open sets containing x . Its * Corresponding Author:Shyamapada Modak, 2017-2019©Published by AksarayUniversity 112 S. Modak & Md.M. Islam (2019). Aksaray University Journal of Science and Engineering, 3(2), 112-128. complement function, that is,  -operator is defined as:  ( A)  X (X A)* . Using these two set functions, ()* and  , Modak and Islam [19, 20] have introduced two moreoperators in the ideal topological spaces and they are: * ( A)  ( A* )  X (X A* )* and * ( A)  (( A))*  {x  X : U x ( A)  } , where U x  ( x) . Following example shows that the values of the operators  * and * are not the same: Example 1.1. Let X  {a, b, c} ,   {,{c}, X } and   ({a, b})  X  {,{c}} . Then, * ( X )   ( X * ) ({c})*  X and ( ( X ))*  X *  {a, b} . Therefore, * ( X )  * ( X ) . The value of the operator * is an open set and the value of the operator  * is a closed set. In this paper, we further consider the operators using joint operators  * and * simultaneously and shall define two more operators using of  * and * which is  and meet of  * and * which is  . We also consider the values of these two operators on various ideal topological spaces as well as various subsets of the ideal topological space. We also give a bunch of characterization of Hayashi-Samuel space. An ideal topological space ( X , , ) is called Hayashi-Samuel space [21], if   ´  {} . Theauthors Hamlett and Jankovi c [3] called it by the name of  -boundary, whereas the authors Dontchev, Ganster and Rose [22] called it by the name of codense ideal. In the study of ideal topological spaces, it played an important role. Two well known Hayashi-Samuel spaces are: Let  be a topology on a set X , then ( X , ,{}) is a Hayashi-Samuel space and if ( X , , n n is the collection of all nowhere dense subsets of ( X , ) , then ) is also a Hayashi-Samuel space. Further, we also give the topological properties of the generalized closure spaces [23, 24] induced by the above mentioned operators  and  . Now we shall give a few words about generalized closure spaces. The study of closure spaces was introduced by Habil and Elzenati [23] in 2003 and Stadler [24] in 2005. Generalized closure space is the generalization of closure space and its definition is as follows: Definition 1.2. Let X be a set, ( X ) be the power set of X and cl :( X ) ( X ) be any arbitrary set-valued set-function, called a closure function. We call cl ( A) the closure of A , and we call the pair ( X , cl ) a generalized closure space (see [23, 24]). Aksaray J. Sci. Eng. 3:2 (2019) 112-128 113 S. Modak & Md.M. Islam (2019). Aksaray University Journal of Science and Engineering, 3(2), 112-128. Consider the following axioms (see [23, 24]) of the closure function for all A, B, A ( X ) ,  is an index set: The closure function in a generalized closure space ( X , cl ) is called: (K0) grounded, if cl ()   . (K1) isotonic, if A  B implies cl ( A)  cl ( B ) . (K2) expanding, if A  cl ( A) . (K3) sub-additive, if cl ( A  B)  cl ( A)  cl ( B) . (K4) idempotent, if cl (cl ( A))  cl ( A) . (K5) additive, if  cl ( A )  cl (  (A )) . Definition 1.3. [24, 25, 26] A pair ( X , cl ) is said to be an isotonic space if it satisfies the axioms (K0) and (K1). If an isotonic space ( X , cl ) satisfies (K2), then it is called a neighbourhood space. A closure space that satisfies (K4), is called a neighbourhood space. A topological space, that satisfies (K3), is a closure space. ‘int ’ is the complement function of the closure function ‘cl’ and it is defined as: int( A)  X \ cl ( X \ A) , for A  X . 2.  Operator Definition 2.1. Let ( X , , ) be an ideal topological space. We define the operator  :( X ) ( X ) as: ( A)  * ( A)  * ( A) , for A  X . Observe that, for A  X , ( A) is the union of an open set and a closed set. The next example shows that union of an open set and a closed set is not always an expression of ( A) , for any A  X . Example 2.2. Let X  {a, b, c} ,   {,{a},{a, b}, X } and  {,{b}} . Let A1  {a} and A2  {c} . Then, A1 is open and A2 is closed. Then A1  A2  {a, c} . Now (())*    ( ({b}))*  ( ({c}))*  ( ({b, c}))* , (({a}))*  X  (({a, b}))*  ( ({a, c}))* Aksaray J. Sci. Eng. 3:2 (2019) 112-128 114 S. Modak & Md.M. Islam (2019). Aksaray University Journal of Science and Engineering, 3(2), 112-128.  ( ( X ))* and (* )     (({b})* )   (({c})* )  (({b, c})* ) ,  (({a})* )  X   (({a, b})* )  (({a, c})* )  ( X * ) . So there is no T ( X ) such that (T )  A1  A2 . If  {} , then ( A)  Int (Cl ( A))  Cl ( Int ( A)) (where ‘Int’ and ‘Cl’ denote the interior and closure operator of ( X , ) respectively) and if  n , then ( A)  [ Int (Cl ( Int (Cl ( Int (Cl ( A))))))]  [Cl ( Int (Cl ( Int (Cl ( Int ( A))))))]  Int (Cl ( A))  Cl ( Int ( A)) . Therefore, the value of  , for any subset A of X on ( X , ,{}) and ( X , , n ) are equal. The operator  is not grounded and it follows from the following example: Example 2.3. Let X  {a, b, c, d } ,   {,{a}, X } and  {,{a}} . Then, ()  * ()  * ()    {a}  {a}   . So, the operator  is not grounded. Theorem 2.4. An ideal top (...truncated)