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.
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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.
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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.
Remark:3.2.In an ideal space (X,, ), every -closed set is an RIC-open whereas every open set is an RIC -closed set.
Definition3.3.Let A be a subset of an ideal topological space (X,, ). We define RIC-interior of the set A is the largest RIC-open set contained in A.We will denote RIC-interior of a set A by RIC-int(A).
Definition3.4.Let A be a subset of an ideal topological space (X,, ).A point xA is said to be an RIC-interior point of the set A if there exists an RIC-open set U of x such that xU A.
Definition 3.5.Let (X, , ) be an ideal space and xX . We define R IC - neighborhood of x as an R IC -open set containing x. We denote the set of all R IC - neighborhoods of x by R IC -N(x).
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Definition3.6.Let A be a subset of an ideal topological space (X,, ). We define R IC -closure of the set A as the smallest R IC -closed set containing A. We will denote R IC -closure of a set A by R IC-cl(A).
* Proposition3.7.In an ideal space (X, , ), R IC -topology is coarser than Rc -topology.
* * Proof: Since every R-perfect is an R -perfect set, R IC -topology is coarser than Rc -topology. Example3.8.Let X = {a,b,c,d}, = {,{a},{b},{a,b},{a,d},{a,b,d},X} and I= {,{b},{c},{b,c}}. The set { ,{c},{d},{c,d}{a,d},{b,d},{b,c},{a,b,d},{b,c,d},{a,c,d},X} is RIC –topology. * The set { ,{b},{c},{d},{c,d}{a,d},{b,d},{b,c},{a,b,d},{b,c,d},{a,c,d},X} is Rc –topology
Remark 3.8.(i) Since every open set is an RIC -open set,every neighborhood U of a point x X isan RIC-neighborhood of x. (ii) If x X is an interior point of a subset A of X, then x is an RIC-interior point of A.
(iii) From (ii),we have int(A) RIC-int(A) and int(A) = RIC-int(A) when = {}.
Theorem:3.9.Let A and B be subsets of an ideal space (X,, ) .Then the following properties hold:
(i)RIC-int(A) = {U: U A and U is an RIC -open set}. (ii)RIC-int(A) is the largest RIC -open set of X contained in A. (iii)A is RIC -open if and only if then A = RIC-int(A). (iv)RIC-int (RIC-int(A)) = RIC-int(A) . (v)If A B, then RIC-int(A) RIC-int(B)
Definition 3.10.Let A be a subset of an ideal topological space (X, , ) We define RIC-closure of the set A as the smallest RIC-closed set containing A. We will denote RIC -closure of a set A by RIC-cl(A).
Remark 3.11.For any subset A of an ideal topological space (X, , ) , RIC-cl(A) cl(A). If ={}thenRIC-cl(A) = cl(A) .
Theorem 3.12.Let A and B be subsets of an ideal space (X, , ) with X is finite. Then thefollowing properties hold: (i)RIC-cl(A) = {F : A F and F is RIC -closed set} (ii)A is RIC -closed if and only if A = RIC-cl(A) (iii)RIC-cl(RIC -cl(A)) = RIC -cl(A) (iv)If A B, then RIC-cl(A) RIC-cl(B).
Theorem 3.13.Let A be a subset of an ideal space (X, , ).Then the following properties hold: (i)RIC-int(X-A) = X \ (RIC-cl(A)). (ii)RIC-cl (X-A) = X \ (RIC-int (A)).
Definition3.14.Let (X, , ) be an ideal space with X is finite. The collection of all RIC-closed sets is a topology
which is denoted by RI( ,). If there is no confusion about the topology and the ideal , we will call the
topological space RI ( ,) as RI- The members of RI( ,)are calledas RI-open sets and the complements of RI- open sets are called as RI -closed sets.
Definition3.15.Let A be a subset of an ideal topological space (X,, ). We define RI-interior of the set A is the largest RIC-open set contained in A. We will denote RI-interior of a set A by RI-int(A). A point xA is said to be an RIC-interior point of the set A if there exists an RIC-open set U of x such that xU A.We define R ICneighborhood of x as an R IC -open set containing x. We denote the set of all R IC - neighborhoods of x by R IC-N(x).Also R I - closure of the set A as the smallest R I-closed set containing A. We will denote R IC-closure of a set A by R IC-cl(A).
* Remark 3.16.Every open set is RI-open and every RI-open set is R -open.
c Remark 3.17. In an ideal space (X, , ), a subset A of X is RI-open set if and only if A is an RIC-open set. In c other words, a subset A of X is RIC-closed set if and only if A is an RI-open set.
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Example 3.18.(forR I-topology)Let X = {a,b,c,d}, = {,{a,b},X} and I= {,{a}}. Then the set {,{c,d}{b,c,d},X}is R I-topology.
*. Proposition3.19. In an ideal space (X, , ), RI R
* Remark 3.20.The following example shows that R ≠RI ≠ .
Example 3.21.Let X = {a,b,c,d}, = {,{a},{a,b,c},X} and I= {,{a},{b},{a,b}}.
Then the collection {,{a},{b},{a,b},{a,b,c},X} is RI–topology. The collection{,{a},{b},{c},{a,b},{b,c},{a,c},{c,d},{a,b,c},{b,c,d},{a,c,d},X} is R*–topology.
Remark:3.22We observe that * R*.
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