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New Sets and Topologies in Ideal Topological Spaces J International Journal of Pure and Applied Mathematics Volume 118 No. 20 2018, 93-98 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu NEW SETS AND TOPOLOGIES IN IDEAL TOPOLOGICAL SPACES J.Sebastian Lawrence Department of Mathematics Sathyabama University,Chennai-600119 Email: [email protected] R.Manoharan, Department of Mathematics, Sathyabama University, Chennai-600119. Email: [email protected] . ABSTRACT In this paper, in an ideal space(X, , ), we introduce and study about a new topology called RIC - topology which is finer than .Also we analyze the relationship with the already existing topologies . AMS Subject classification: 54A05,54A10 Keywords- R -perfect sets, R -open sets,R -open sets, R –topology. I IC I IC 1. Introduction A nonempty collection of subsets of a set X is said to be an ideal on X, if it satisfies the following two conditions: (i) if A and B A , then B (heredity); (ii) if A and B ,thenA B (finite additivity) . An ideal topological space ( or ideal space) (X, , ) means a topological space (X, ) with an ideal defined on X. For a given point x in a space ( X, ), the system of open neighborhoods of x is denoted by N(x) ={U : x U}. Let (X, ) be a topological space with an ideal defined on X. then for any subset A of X, A* ( , ) ={x X/A U for every U N(x)} is called the local function of A with respect to and . If there is no ambiguity, we will write A*( ) or simply A* for A*( , ) . Also, cl*(A) = A A* defines a Kuratowski closure operator[3] for the topology *( ) ( or simply *) which is finer than .Jankovic and Hamlett[1,3,4] made an extensive study on topological ideals which initiated the generalization of many topological properties. Manoharanet al. [7] introduced R*-topology which is finer than *. Let (X,, ) be an ideal topological space. A subset A of X is said to R*-perfect if A*\ A . The collection of all perfect setssatisfies the conditions of being a * * * basis for some topology and it willbe called as R c(, ). we define R (, )={AX / X \ A R c(,, )} on a nonempty set .Clearly, R* (, ) is a topology if the set X is finite. The members of the collection R*(, )will be called R*-open sets. If there is no confusion about the topology and the ideal , then R* (, ) can be called as R*-topology when X is finite. For a given subset A of a space ( X, ) , cl(A) and int(A) are used to denote the closure of A and interior of A, respectively, with respect to the topology. A perfect set in topological spaces is a set without isolated points (dense in itself) and closed. Hayashi introduced *- perfect sets[2] in ideal topological spaces. A subset A of an ideal space is *-perfect if A = A*. Later, Manoharanet al.[7] introduced R*-perfect sets . Muhammad Shabiret al.[8] introduced and studied soft topological spaces. Kandilet al.[6] extended soft topological properties to soft ideal topological spaces. Recently, Rodyna A. Hosny[9] extended the idea of perfect sets to soft ideal topological spaces. In this chapter, we introduce LI-perfect sets, RI- perfect sets and CI-perfect sets, study their properties and obtain some relation with other sets in ideal topological spaces which also can be extended to soft ideal topological spaces. 93 International Journal of Pure and Applied Mathematics Special Issue 2. LI-Perfect , RI-Perfect and CI-perfect sets In this section, we define three collections of subsets LI, RI and CIin an ideal space and study some of their properties. Definition 2.1.Let (X, , ) be an ideal topological space. A subset A of X is said to be ’ (i) LI-perfect if A \ A ’ (ii) RI-perfect if A \ A and (iii) CI-perfect if A is both LI-perfect and RI-perfect. The collection of LI-perfect sets, RI-perfect sets and CI-perfect sets in (X, , ) are denoted byLI, RI and CIrespectively. Remark 2.2.If = {}, then LIis the collection of all dense-in itself sets,RIis the collection of all closed sets and CIis the collection of all perfect sets. Remark2.3.Every perfect set is both RI-perfect and LI-perfect (and hence CI-perfect).The following examples shows that every RI-perfect and LI-perfect sets need not be a perfect set. Example 2.4.Let X={a,b,c,d}, = {,{a},{b},{a,b},{a,d},{a,b,d},X} and I= {,{b},{c},{b,c}}. The sets {b} and{c} are RI-perfect and LI-perfect sets but not perfect sets. Proposition2.5.Let (X, , ) be an ideal space. Let A be a subset of X. * (i)If A is L -perfect, then it is LI-perfect. * (ii)If A is RI-perfect, then it is R -perfect. The following example shows that the converse of the above results need not be true. Example 2.6.Let X = {a,b,c,d}, = {,{c},{a,b},{a,b,c},X} and I= {,{a}}.The set {b,d} * * is LI-perfect but not L perfect set. The set{a,d}is R -perfect but not RI-perfect. Proposition 2.7.Let (X, , ) be an ideal space. Let A and B are two subsets of X. (i)If A and B are RI- perfect sets, then AB is an RI-perfect set. (ii)If A and B are LI- perfect sets, then AB is an LI- perfect set. Proof. Let A and B are R-perfect sets. Then A’\ AI and B’ \ B . By finite additive property of ideals, (A’\ A) (B’\ B) . Since (A’ B’) \ (A B) (A’\ A) (B’ \ B), by heredity property, (A’ B’) \ (A B) . Hence (A B)’ \ (A B) . This proves (i). Let A and B are LI-perfect sets. Since A and B are LI- perfect sets, A \ A’ and B \ B’ . Hence, by finite additive property of ideals, (A \ A’) (B \ B’) . Since (A B) \ (A B)’ = (A B) \ (A’ B’) (A \ A’) (B \ B’), by heredity property, (A B) \ (A B)’ This proves AB is an LI- perfect set. Corollary2.8.In an ideal space (X, , ), (i)Finite union of RI-perfect sets is an RI-perfect set, (ii)Finite union of LI-perfect sets is an LI-perfect sets. Proposition 2.9.If A and B are R1-perfect sets, then A B is an RI-perfect set. Proof. Suppose A and B are R-perfect sets. Then A’\ AI and B’\ BI. By the finite additive property of ideals, (A ’ \ A) (B’ \ B) . Since (A’ B’) \ (A B) (A’\ A) (B’ \ B) , by heredity property (A’ B’) \ (A B) . Also (A B)’ \ (A B) (A’ B’) \ (A B) . This proves the result. Corollary 2.10.Finite intersection of RI- perfect sets is an RI-perfect set. 94 International Journal of Pure and Applied Mathematics Special Issue Proposition 2.11.Let A be a subset of an ideal space (X, , ).Then (i)If A is C-perfect, then A A’ . (ii)If A is - closed, then A is RI-perfect. (iii)If A is dense-in-itself, then A is LI-perfect. * Corollary 2.12. For any set A, A and cl(A) are RI-perfect sets. * Corollary:2.13.If a set A is an RI-perfect set, then cl (A) is also an RI-perfect. Proof. * * * * Since cl A= AA and both A and A are RI-perfect sets, cl (A) is also an RI-perfect set. * Corollary 2.14. If A is an RI-perfect set, then A A is also an RI-perfect. Proposition 2.15.Let (X, , ) be an ideal space. Let A and B are two subsets of X such that A B and A’ = B’. Then (i) B is RI-perfect if A is RI-perfect. (ii) A is LI-perfect if B is LI-perfect. Proof. * ’ Let A is RI-perfect. Then A’\ AI. Now, B’ \ B = A \ B A \ A. By heredity property of ideals, B’ \ B . Then B is R1- perfect. Let B is LI- perfect. Then B \ B’ . Now, A \ A’ = A \ B’ B \ B’.By heredity property of ideals , A \ A’ . Hence A is LI- perfect. Proposition 2.16.Let (X, , ) be an ideal topological space with X is finite. Then the collection RIof all RI-perfect sets is a topology which is finer than the collection of all -closed sets. Proof. X and are R-perfect sets. By Corollary 2.8.finite union of RI-perfect sets is an RI-perfect set and by Corollary 2.10, finite intersection of RI-perfect sets is RI-perfect. Hence the collection RI is a topology if X is finite. Also, by Proposition 2.11 every -closed set is an RI-perfect set. Hence the topology RI is finer than the collection of all - closed sets if X is finite. 3. RIC-Topology By Proposition 2.9 and 2.11, we observe that the collection of all RI-perfect sets RI={A X / A’ \ A }satisfies the conditions of being a basis for some topology . We will denote the topology as RIC( ,). If there is no confusion about the topology and the ideal , we will call the topological space RIC ( ,) as RIC. The members of RIC( ,)are calledas RIC-open sets and the complements of RIC-open sets are called as RIC -closed sets. Example:3.1.LetX={a,b,c,d},={,{d},{a,c},{a,c,d},X}and I={,{c},{d},{c,d}} then { ,{b},{a,b},{b,d},{a,b,c},{a,b,d},X} is RIC –topology.
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