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Asian Journal of Current Engineering and Maths 2 : 5 September – October (2013) 312 - 315. Contents lists available at www.innovativejournal.in ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Journal homepage: http://www.innovativejournal.in/index.php/ajcem REGULAR PRE SEMI I LOCALLY CLOSED SETS IN IDEAL TOPOLOGICAL SPACES B. Sakthi Sathya*, S. Murugesan Department of Mathematics, G. Venkataswamy Naidu College, Kovilpatti. Tamilnadu. India. Department of Mathematics, S. Ramasamy Naidu Memorial College, Sattur. Tamilnadu. India. ARTICLE INFO ABSTRACT Corresponding Author B. Sakthi Sathya Department of Mathematics, Department of Mathematics, G. Venkataswamy Naidu College, Kovilpatti. Tamilnadu. India. Key Words: rpsI-open, rpsI-closed, rpsIlc-sets, rpsIlc*-sets, rpsIlc**-sets, locally rpsI-connected. In this paper, we introduce and study the new class of sets called regular pre semi I locally closed sets (briefly. rpsIlc-closed) in ideal topological spaces and established their relationships with some generalized closed sets in ideal topological spaces. Also we define and study the notions of RPSILC-continuity and locally rpsI-connectedness. AMS Subject Classification: 54A05. ©2013, AJCEM, All Right Reserved. 1. INTRODUCTION Ideals in topological spaces have been considered since 1930. This topic has won its importance by the paper of Vaidyanathaswamy[8]. A subset A of a space (X,) is locally closed [1] if A is the intersection of an open set and a closed set. Locally closed sets are further investigated by Ganster and Reilly in [3]. In 1999, Dontchev[2] introduced I-locally closed subsets in an ideal topological spaces. In this paper we introduce new class of sets namely rpsIlc-set, rpsIlc*-set and rpsIlc**-set and discuss about their properties. Finally we use the concepts of rpsIlc-set, rpsIlc*-set and rpsIlc**-set to define RPSILC-continuous, RPSILC*-continuous, RPSILC**-continuous functions also we define locally rpsI-connected space and discuss about their relationships with some other connected spaces. 2. PRELIMINARIES Definition 2.1 An ideal is defined as a non-empty collection I of subsets of X satisfying the following two conditions. i. If A I and B A, then B I ii. If A I and B I then A B I An ideal topological space is a topological space (X, ) with an ideal I on X and it is denoted by (X, , I). For A X, A*(I,) = {x X / A U I, for every U (x)}, where (x) = {U / x U}. Note that cl*(A) = A A* defines a Kuratowski operator for a topology *( I ) (also denoted by * when there is no chance for confusion), finer than . A basis β(I, ) = {U \ I : U and I I}. β is not always a topology[5]. cl*(A) and int*(A) will denote the closure and interior of A in (X, *) respectively. Definition 2.2 A subset A of an ideal topological space (X, , I) is called i. regular generalized I-closed[6](rgI-closed) if cl*(A) U whenever A U and U is regular I-open. ii. pre generalized pre regular I-closed[6](pgprIclosed) if pIcl(A) U whenever A U and U is regular generalized I-open. iii. regular pre semi I-closed[6](rpsI-closed) if spIcl(A) U whenever A U and U is regular generalized I-open. The complement of the above mentioned generalized Iclosed sets are their respective I-open sets. rpsIcl(A) is the smallest rpsI-closed set containing A. Definition 2.3 A subset A of an ideal topological space (X, , I) is called a) *-perfect[4] if A = A* b) I-locally closed[2] if A = GV, where G is open and V is *-perfect. We will denote the collection of all I-locally closed sets in (X, , I) by ILC(X,). 3.RPSILC*-SETS AND RPSILC**-SETS In this section, we introduce rpsIlc-set, rpsIlc*-set and rpsIlc**-set each of which is stronger than rpsIlc-set and is weaker than I-locally closed set and study their relations with existing ones. Definition 3.1 A subset A of an ideal topological space (X, , I) is called a) rpsIlc-set if there exists a rpsI-open set U and a rpsI-closed set F of X such that A = U F. b) rpsIlc*-set if there exists a rpsI-open set U and a closed set F of X such that A = U F. c) rpsIlc**-set if there exists an open set U and a rpsIclosed set F of X such that A = U F. The collection of all rpsIlc-sets(resp. rpsIlc*-sets, rpsIlc**sets) of (X, , I) will be denoted by RPSILC(X, ) (resp. RPSILC*(X, ), RPSILC**(X, )). From the above definitions we have the following results. Theorem 3.2 312 Sakthi et. al/ Regular Pre Semi I Locally closed Sets In Ideal Topological Spaces For an ideal topological space (X, , I) the following implications hold: a) ILC(X,) RPSILC*(X, ) RPSILC(X, ) b) ILC(X,) RPSILC**(X, ) RPSILC(X, ) The reverse implications need not be true as seen from the following example. Example 3.3 Let X = {a,b,c,d} = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {, {a}}. Then we have ILC(X,) = {, X, {a}, {b}, {c}, {d}, {a,b}, {b,c}, {c,d}, {a,b,c}, {b,c,d}}. RPSILC(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. In this ideal space the set {a,c} is rpsIlc-set but not Ilc-set. RPSILC*(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. The set {a,d} is rpsIlc*-set but not Ilc-set. Remark 3.4 RPSILC* and RPSILC** are independent of each other. Example 3.5 Let X = {a,b,c,d} = {, X, {a,c}, {d}, {a,c,d}} and I = {, {c},{d},{c,d}}. Then we have RPSILC(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. RPSILC*(X,) = {, X, {a}, {b}, {d}, {a,b}, {a,c}, {a,d}, {b,d}, {a,b,c}, {a,b,d}, {a,c,d}}. RPSILC**(X,) = {, X, {b}, {c}, {d}, {a,c}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,c,d}, {b,c,d}}.In this ideal space the set {c} is rpsIlc**-set but not rpsIlc*-set and the set {a,b} is rpsIlc*set but not rpsIlc**-set. Also the set {b,c} is rpsIlc-set but not rpsIlc*-set and the set {a} is rpsIlc-set but not rpsIlc**set. From Theorem 3.2 to Example 3.5 we have the following diagram. Theorem 3.6 Let A and B be subsets of (X, , I). If A RPSILC(X,) and B RPSILC(X,) then AB and AB RPSILC(X,). Proof: The proof follows from Definition 3.1 Remark 3.7 The union and intersection of two rpsIlc*-sets need not be a rpsIlc*-set. Example 3.8 Let X = {a,b,c,d} = {, X, {a}, {a,b}, {b,d}, {a,b,d}} and I = {, {a}}. Then we have RPSILC*(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {b,d}, {b,c}, {c,d}, {a,b,c}, {a,b,d}, {b,c,d}}. In this ideal space {a}{c,d}= {a,c,d} which is not rpsIlc*-set. Example 3.9 Let X = {a,b,c,d} = {, X, {a,b,c}} and I = {, {a}}. Then we have RPSILC*(X, ) = {, X, {b}, {c}, {d}, {a,b}, {a,c}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. In this ideal space {a,b} {a,c}={a} which is not rpsIlc*-set. Remark 3.10 The union and intersection of two rpsIlc**-sets need not be a rpsIlc**-set. Example 3.11 Let X = {a,b,c,d} = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {, {a}}. Then we have RPSILC**(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. In this ideal space {a,b,d} {b,c,d}={b,d}, {b}{d}= {b,d} which are not rpsIlc**-set. Theorem 3.12 Let A be any subset of X, a) A is rpsI closed in X if and only if A = rpsIcl(A) b) rpsIcl(A) is rpsI closed in X c) x rpsIcl(A) if and only if A U for every rpsI-open set U containing x. Proof: (a) and (b) are trivially true. (c). Suppose that there exists a rpsI-open set U containining x such that A U = . Since X\U is rpsI-closed and A X\U, rpsIcl(A) X\U hence x rpsIcl(A). Conversely, suppose that x rpsIcl(A). Then U = X\ rpsIcl(A) is rpsI-open set containing x and A U = . Theorem 3.13 For a subset A of (X, , I), the following statements are equivalent. a) A RPSILC(X,). b) A = U rpsIcl(A) for some rpsI-open set U. c) rpsIcl(A)\A is rpsI-closed. d) A (X\ rpsIcl(A)) is rpsI-open. Proof: (a) (b). Suppose A RPSILC(X,). Then there exists a rpsI-open subset U and rpsIclosed subset F such that A = U F. Since A U and A rpsIcl(A), A U rpsIcl(A). Conversely by Theorem 3.12(b) rpsIcl(A) F and hence U rpsIcl(A) U F = A. Therefore A = U rpsIcl(A). (b) (a). By Theorem 3.12(b) rpsIcl(A) is rpsI closed and hence A = U rpsIcl(A) RPSILC(X,). (c) (d). Let S = rpsIcl(A)\A. Then by the assumption S is rpsI-closed which implies X\S is rpsI-open and X/S = X Sc = X (rpsIcl(A)\A)c = A (X\rpsIcl(A)). Thus A (X\ rpsIcl(A)) is rpsI-open. (d) (c). Let W = A (X\rpsIcl(A)). Then W is rpsIopen. This implies that X\W is rpsI-closed and X\W = X \ A (X\ rpsIcl(A)) = rpsIcl(A) X\A = rpsIcl(A)\A. Thus rpsIcl(A)/A is rpsI-closed. (d) (b). Let U = A (X\ rpsIcl(A)). Then U is rpsIopen. Now U rpsIcl(A) = (A (X\rpsIcl(A))) rpsIcl(A) = (rpsIcl(A) A) (rpsIcl(A) X\rpsIcl(A)) = A = A. Therefore A = U rpsIcl(A) for some rpsI-open set U. (b) (d). Let A = U rpsIcl(A), for some rpsI-open set U. A (X\ rpsIcl(A)) = (U rpsIcl(A)) (X\ rpsIcl(A)) = U (rpsIcl(A) (X\ rpsIcl(A))) = U X = U, is rpsI-open. Theorem 3.14 For a subset A of (X, , I), the following statements are equivalent. a) A RPSILC*(X,). b) A = U cl(A) for some rpsI-open set U. c) cl(A)\A is rpsI-closed. d) A (X\cl(A)) is rpsI-open. Proof: The proof is similar to that of Theorem 3.13 Theorem 3.15 Let A be a subset of (X, , I). Then A RPSILC**(X,) if and only if A = U rpsIcl(A) for some open set U. Proof: 313 Sakthi et. al/ Regular Pre Semi I Locally closed Sets In Ideal Topological Spaces Let A RPSILC**(X,). Then A = U F where U is open and F is rpsI-closed. Since A F implies rpsIcl(A) F. Now A = A rpsIcl(A) = (U F) rpsIcl(A) = U rpsIcl(A). The converse part is obvious. Theorem 3.16 Let A be a subset of (X, , I). If A RPSILC**(X,) then rpsIcl(A)\A is rpsI-closed and A (X\ rpsIcl(A)) is rpsI-open. Proof Suppose A RPSILC**(X,) then by Theorem 3.15 there exists a open set U such that A = U rpsIcl(A). A (X\ rpsIcl(A)) = (U rpsIcl(A)) (X\ rpsIcl(A)) = U (rpsIcl(A) (X\ rpsIcl(A))) = U X = U, which is open. Since every open set is rpsI-open we have A (X\ rpsIcl(A)) is rpsI-open. Let W = A (X\ rpsIcl(A)). Then W is rpsI-open. This implies that X\W is rpsI-closed and X\W = X \ A (X\ rpsIcl(A)) = rpsIcl(A) X\A = rpsIcl(A)\A. Thus rpsIcl(A)\A is rpsI-closed. The converse of the above theorem need not be true as seen from the following example. Example 3.17 Let X = {a,b,c,d} = {, X, {a,b,c}} and I = {, {a}}. Then we have {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,d}, {c,d}, {a,b,d}, {a,c,d}} is the set of all rpsI-closed sets in X and RPSILC**(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}}. If A = {b, c} then rpsIcl(A)\A = {a,d} is rpsI-closed and A (X\ rpsIcl(A) = {b,c} is rpsI-open but A RPSILC**(X,). 4. RPSILC*-continuous and RPSILC**-continuous In this section we define RPSILC*-continuous and RPSILC**-continuous functions in ideal topological spaces and discuss about their relations. Definition 4.1 A function f : (X,,I) (Y,) is called ILC-continuous[2] if f 1(A) is Ilc-set in X, for every open subset A of Y. Definition 4.2 a) A function f : (X,,I) (Y,) is called RPSILC-continuous if f -1(A) is rpsIlc-set in X, for every open subset A of Y. b) A function f : (X,,I) (Y,) is called RPSILC*-continuous if f -1(A) is rpsIlc*-set in X, for every open subset A of Y. c) A function f : (X,,I) (Y,) is called RPSILC**continuous if f -1(A) is rpsIlc**-set in X, for every open subset A of Y. Theorem 4.3 Let f:(X,,I) (Y,) be a function. a) If f is ILC-continuous, then f is RPSILC*-continuous. b) If f is RPSILC*-continuous, then f is RPSILC-continuous. c) If f is RPSILC**-continuous, then f is RPSILC-continuous. d) If f is ILC-continuous, then f is RPSILC-continuous. Proof: The proof follows from Theorem 3.2, Converse of the above theorem need not be true as seen from the following example. Example 4.4 Let X = Y = {a,b,c,d}. = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {, {a}}, = {, Y, {a,c}, {d}, {a,c,d}}. Define f : (X,, I) (Y,) by f(a) = d, f(b) = c, f(c) = b, f(d) = d. Then f is RPSILC*-continuous but not ILC-continuous. Example 4.5 Let X = Y = {a,b,c,d}. = {, X, {a,c}, {d}, {a,c,d}}. and I = {, {c},{d},{c,d}}, = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}}. Define f : (X,, I) (Y,) by f(a) = d, f(b) = b, f(c) = b, f(d) = c. Then f is RPSILC-continuous but not RPSILC*continuous. Example 4.6 Let X = Y = {a,b,c,d}. = {, X, {a,c}, {d}, {a,c,d}}. and I = {, {c},{d},{c,d}}, = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}}. Define f : (X,, I) (Y,) by f(a) = a, f(b) = c, f(c) = f(d) = d. Then f is RPSILC-continuous but not RPSILC**-continuous. Example 4.7 Let X = Y = {a,b,c,d}. = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {, {a}}, = {, Y, {a,c}, {d}, {a,c,d}}. Define f : (X,, I) (Y,) by f(a) = f(c) = d, f(b) = b, f(d) = c. Then f is RPSILC-continuous but not ILC-continuous. Remark 4.8 RPSILC*-continuous and RPSILC**-continuous functions are independent of each other. Example 4.9 Let X = Y = {a,b,c,d}. = {, X, {a,c}, {d}, {a,c,d}}. and I = {, {c},{d},{c,d}}, = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}}. a) Define f : (X,, I) (Y,) by f(a) = d, f(b) = c, f(c) = a, f(d) = c. Then f is RPSILC**-continuous but not RPSILC*-continuous. b) Define f : (X,, I) (Y,) by f(a) = b, f(b) = b, f(c) = d, f(d) = c. Then f is RPSILC*-continuous but not RPSILC**-continuous. Theorem 4.3 to Example 4.9 we have the following Diagram 5. RPSILC-irresolute Functions Definition 5.1 a) A function f : (X,,I) (Y,) is called RPSILC-irresolute if f -1(A) is rpsIlc-set in X, for every rpsIlc-set A of Y. b) A function f : (X,,I) (Y,) is called RPSILC*-irresolute if f -1(A) is rpsIlc*-set in X, for every rpsIlc*-set A of Y. c) A function f : (X,,I) (Y,) is called RPSILC**-irresolute if f -1(A) is rpsIlc**-set in X, for every rpsIlc**-set A of Y. Theorem 5.2 Let f:(X,,I) (Y,) be a function. a) If f is RPSILC* irresolute function then f is RPSILCirresolute function. b) If f is RPSILC**-irresolute function, then f is RPSILCirresolute function. Proof: The proof follows from Theorem 3.2 The converse of the above theorem need not be true as seen from the following example. Example 5.3 Let X = Y = {a,b,c,d}. = {, X, {a,c}, {d}, {a,c,d}}. and I = {, {c},{d},{c,d}}, = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and J = {,{a}}. Define f : (X,, I) (Y,, J) by f(a) = a, f(b) = b, f(c) = c, f(d) = d. Then f is RPSILC-irresolute but not RPSILC*irresolute and RPSILC**-irresolute. Remark 5.4 RPSILC*-irresolute and RPSILC**-irresolute functions are independent of each other. Example 5.5 Let X = Y = {a,b,c,d}. = {, X, {a}, {a,b}, {b,d}, {a,b,d}}, I = {, {a}}, = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and J = {,{a}}. 314 Sakthi et. al/ Regular Pre Semi I Locally closed Sets In Ideal Topological Spaces a) Define f : (X,,I) (Y,,J) by f(a) = b, f(b) = a, f(c) = a, f(d) = d. Then f is RPSILC*-irresolute but not RPSILC**irresolute. b) Define f : (X,, I) (Y,, J) by f(a) = b, f(b) = c, f(c) = d, f(d) = b. Then f is RPSILC**-irresolute but not RPSILC*irresolute. Theorem 5.2 to Example 5.5 we have the following Diagram. 6 Locally rpsI-connected Spaces: In this section the idea of locally rpsI-connectedness in an ideal topological space is introduced and obtain some of its basic properties. Definition 6.1 An ideal topological space (X,, I) is called semiIconnected if it can not be written as the union of two disjoint non empty semiI open sets. Definition 6.2 An ideal topological space (X,, I) is called rpsIconnected[7] if it can not be written as the union of two disjoint non empty rpsI-open sets. Definition 6.3 An ideal topological space (X,, I) is called locally connected at x X if for every open set U containing x, there exists a connected, open set C such that x C U. An ideal topological space (X,, I) is called locally connected if it is locally connected at every point of X. Definition 6.4 An ideal topological space (X,, I) is called locally rpsI-connected at x X if for every rpsI-open set E containing x, there exists a rpsI-connected, open set C such that x C E. An ideal topological space (X,, I) is called locally rpsI-connected if it is locally rpsI-connected at every point of X. Lemma 6.5[7] Every rpsI-connected space is connected. Remark 6.6 Every locally rpsI-connected space is locally connected but the converse is not true as seen from the following example, consider the ideal topological space (X,, I), where X = {a,b,c,d}, = {, X, {a}, {b,d}, {a,b,d}}. and I = {, {a}}. rpsI-open sets of X are , X, {a}, {b}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d} and rpsI-closed sets are , X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,c,d}, {b,c,d}.Here X is locally connected but not locally rpsI-connected because the sets X, {b,c,d}, {a,b,d}, {a,b,c}, {b,d}, {b,c}, {a,b}, {b} are rpsI-open sets containing b, but there is no open subset of {b} containing b and so X is not locally rpsI-connected. Remark 6.7 Locally rpsI-connected and rpsI-connected spaces are independent of each other. For example, let X ={a,b,c,d}, = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {, {a}}. Then we have {a,b} is locally rpsI-connected but not rpsIconnected and {b,c,d} is rpsI-connected but not locally rpsIconnected. Remark 6.8 Locally rpsI-connected and semiI-connected spaces are independent of each other. For example, consider the ideal topological space in Remark 5.5, {b,c,d} is semiIconnected but not locally rpsI-connected and {a,b} is locally rpsI-connected but not semiI-connected. Theorem 6.9 Every rpsI-connected space is locally-connected. Proof: Proof follows from Lemma 6.5 But the converse need not be true as seen from the following example. Consider X = {a,b,c,d}, = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {, {a}}. Then we have {a,b} is locally connected but not rpsI-connected. From Remark 6.5 to Theorem 6.9 we have the following Diagram REFERENCES [1] N. Bourbaki, General Topology, Part I, Addison-Wesley, Reading, Mass, 1966. [2] J.Dontchev, Idealization of Ganster Reilly Decomposition Theorems, Math.GN/9901017,(5), Jan.1999. [3] M. 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