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1 Research Paper α -SETS IN IDEAL TOPOLOGICAL SPACES 1 K. V. Tamil Selvi, 2 P. Thangaraj, 3 O. Ravi Address for Correspondence 1 Department of Mathematics, Kongu Engineering College, Perundurai, Erode District, Tamil Nadu, India. 2 Department of Computer Science and Engineering, Bannari Amman Institute of Technology, Sathyamangalam, Erode District, Tamil Nadu, India. 3 Department of Mathematics, P.M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India. ABSTRACT The aim of this paper is to introduce α -sets in ideal topological spaces. The relationships between the class of α -sets and the related sets are discussed. KEYWORDS AND PHRASES. - set, α - set, weak - set, t- -set, α*- -set, α- -openset,semi- -regular set, regular- -closed set. 1. INTRODUCTION AND PRELIMINARIES In 1986, Tong [32] introduced -sets and in 1989, Tong [33] introduced -sets in topological spaces. In 1998, Dontchev [7] introduced the class of -sets which lies between the class of -sets and the class of -sets. In 2009, Ekici and Noiri [11] introduced the class of α sets which is weaker form of the class of -sets. They also studied the relationships between α -sets and the related sets; and some decompositions of α -continuity, α- continuity, continuity and -continuity were provided by them. In this paper, the class of α -sets is introduced in ideal topological spaces. Some new relationships between the class of α -sets and the related sets are obtained. Also, properties of α -sets are discussed. In the present paper(x,τ) or(y,σ) will denote topological spaces with no separation properties assumed. For a subset V of X, let cl(v) and int(v) denote the closure and the interior of V, respectively, with respect to the topological space(x,τ). An ideal on a topological space (X, τ) is a non-empty collection of subsets of X which satisfies the following conditions. (1) A and B A imply B and (2) A and B imply A B. Given a topological space (X, τ ) with an ideal on X if (X) is the set of all subsets of X, a set operator ( )* : (X) (X), called a local function [34] of A with respect to τ and is defined as follows: for A X, A* (, τ ) = {x X U A for every U τ(x)} where τ(x)={u τ x U}. A Kuratowski closure operator cl* ( ) for a topology τ* (, τ ), called the * -topology, finer than τ is defined by cl* (A) = A A* (, τ ) [25]. We will simply write A* for A* (, τ ) and τ* for τ* (, τ ). If is an ideal on X, then (X, τ, ) is called an ideal topological space. Notice that int* (A) denotes the interior of A in (X, τ* ). Definition 1.1.A subset H of an ideal topological space (X, τ, ) is called 1. α- -open [16] if H int(cl* (int(h))), 2. semi- -open [16] if H cl* (int(h)), 3. pre- -open [8] if H int(cl* (H)), 4. t- -set[16] if int(cl* (H)) = int(h), 5. an α* - -set[16]if int(h)=int(cl* (int(h))), 6. regular- -closed[22]if H=(int(H))*, 7. * -closed [21] if H* H or cl* (H) = H, 8. semi* - -open[12] if H cl(int * (H)), 9. semi* - -closed[12] if its complement is semi*- -open, 10. β- -open [16] if H cl(int(cl* (H))), Mathematics 11. Subject β- -closed[16] Classification: if its 54A10, complement 54C10, 54D10, is β- -open 54D15

2 12. semi - -regular [23] if H is both a t- -set and a semi -open set, 13. β- -regular [35] if H is both a β-i-open set and an α*- -set, 14. strong β- -open [18] if H cl* (int(cl* (H))), 15. τ*-dense [20] if cl* (H) =X, 16. an α -set [28] or α I N3-set[3] if H α (X) ={A B:A is α- -open and B is at- -set}, 17. an α -set [28] or α N4-set[3] if H α (X)={A B: A is α- -open and B is an α* - -set}, 18. an -set [16] if H (X)={A B:A is open and B is an α* - -set}, 19. an -set [23] if H (X) ={A B : A is open and B is semi- -regular}, 20. weak -set [35] if H W (X)={A B:A is open and B is β- - regular}, 21. weakly- -locally closed [24] if H W LC(X)={A B:A τ and B is* -closed}, 22. an -set [22] if H (X)={A B:A τ and B is regular- -closed}, 23. a -set [16] if H (X)={A B:A τ and B is at- -set}. Proposition 1.2. [23] Every regular- -closed set is a semi- -regular but not conversely. Proposition 1.3. [16] Every t- -set is an α* - -set. Proposition 1.4. [23] Every -se tis a -set. Proposition 1.5. [16] Every open set is an α- -open but not conversely. Proposition 1.6. [2]The following are equivalent for a subset H of an ideal topological space (X, τ, ): (1) α- -open. (2) semi- -open and pre- -open. Proposition 1.7. [18] Every semi-i-open set is a strong β- -open. Proposition 1.8. [23] (1) Every semi- -regular set is at- -set. (2) Every semi-i-regular set is a semi-i-open set. Proposition 1.9. [23] For a subset of an ideal topological space (X, τ, ), the following property holds: Every AB -set is semi- -open. Remark [35]The following properties hold for a subset of an ideal topological space (X, τ, ): (1) Every semi- -regular set is β- -regular but not conversely. (2) Every -set is a weak -set but not conversely. (3) Every weak -set is a -set but not conversely. (4) The notions of α- -open sets and weak -sets are independent. Remark [14] Let H be a subset of an ideal topological space (X, τ, ). Then the β- -closure of H, denoted by β- -cl(h), is the smallest β- -closed set containing H. Remark [22] Every regular- -closed set is * -closed. Remark [29] Let H be a subset of an ideal topological space (X, τ, ). The smallest semi*- closed set containing H is called the semi*- -closure of H and is denoted by s* -cl(h). Theorem [29] Let (X, τ, ) be an ideal topological space and H X. Then the following hold: s* cl(h) =H int (cl* (H)).

3 Theorem [29] Let (X, τ, ) be an ideal topological space and H be a strong β- -open subset of X. Then s* cl(h) is semi- -regular. Lemma1.16. [28]The following are equivalent for a subset H of an ideal topological space (X, τ, ): (1) H is an α -set. (2) H=U s* cl(h) for some α- -open set U Proposition [2] Let H be a subset of an ideal topological space (X, τ, ). (1) If V is semi- -open and A is α- -open,then H=V A is semi- -open. (2) If V is pre- -open and A is α- -open, then H=V A is pre- -open. Lemma [4] Let (X, τ, ) be an ideal topological space and A X. Then A is α- -open if and only if A = U V where U is open and int(v) is τ* -dense. Definition [6] A subset H of an ideal topological space (X, τ, ) is said to be -dense if H* = X. Definition [6] An ideal topological space (X, τ, ) is said to be -hyper connected if every non empty open subset of X is -dense. Theorem [30] Let H be a subset of an ideal topological space (X, τ, ). Then the following holds. H is semi- -regular if and only if H is both strong β- -open and semi* - -closed. Proposition In an ideal topological space (X, τ, ), (1) if U τ and W is α- -open set, then U W is α- -open[26]. (2) if U τ and W is β- -open set, then U W is β- -open[17]. Definition [12] An ideal topological space X is called *-extremally disconnected if the *-closure of every open subset of X is open. Lemma [19] Let K be a subset of an ideal topological space (X, τ, ). If N is open, then N cl * (K) cl* (N K). Definition1.25. An ideal topological space (X, τ, ) is called I-submaximal [1, 13]if every τ*-dense subset of X is open. Theorem1.26. [13]For an ideal topological space(x, τ, ),the following properties are equivalent: (1) X is -submaximal. (2) Every pre- -open set is open. (3) Every pre- -open set is semi- -open and every α- -open set is open. 2. α -sets Definition 2.1. A subset H of an ideal topological space (X, τ, I) is called (1) an α -set if H α (X) ={U V:U is α- -open and V is semi- - regular}. (2) an -set if cl* (int(h))=x. Remark 2.2. (1) Every α- -open (semi- -regular)set is an α -set but not conversely. (2) The following diagram holds for a subset H of an ideal topological space (X, τ, ): -set -set -set -set

4 α -set α - set α -set None of these implications is reversible as shown in the following examples. Example 2.3. Let X = {a, b, c, d}, τ = {, {a},{b},{a, b},{b, c},{a, b, c},x} and I={,{b}}. Then {a, d} is an α -set but not an α- -open. Example 2.4. Let X, τ and be as in Example2.3. Then {a, b, c} is an α -set but not a semi-iregular. Example 2.5. Let X={a, b, c, d},τ={,{a},{d},{a, d},x} and I={,{d}}. Then {a, b} is an -set but not an - set. Example 2.6. Let X={a, b, c, d},τ={,{d},{a, c},{a, c,d},x}and ={,{c},{d},{c, d}}.then {b } is a - set but not an -set. Example 2.7. In Example 2.6,{a, b, d} is a - set but not a - set. Example 2.8. Let X, τ and be as in Example 2.5.Then {a, b, d} is an α -set but not an - set. Example 2.9. Let X={a, b, c, d},τ={,{a},{a, b},x} and ={,{c}}.then {a, b, c} is an α - set but not a - set. Example In Example 2.6, {b} is an α - set but not an α - set. Example Let X, τ and be as in Example2.5.Then {a, b, d } is an α - set but not a - set. Example Let X={a, b, c},τ={,{a, b}, X} and ={,{c}}. Then {b, c} is an α - set but not an α -set. Proposition For a subset of an ideal topological space (X,τ, ),the following property holds: Every α - set is semi- -open. Proof. Let H be an α - set. Then H = U V where U is an α- - open set and V is a semi- -regular set. By Proposition 1.8, V is a semi- -open set. By Proposition 1.17(1), H is semi- open. Theorem 2.14.ThefollowingareequivalentforasubsetHofanidealtopological space (X, τ, ): (1) H is an α - set. (2) H is a semi- -open and an α -set. (3) H is a strong β- -open and an α - set. Proof. (1) (2): Every α - set is a semi- -open and an α - set. (2) (3):Obvious. (3) (1):Let H be a strong β- -open and an α - set. By Lemma1.16, H= U s* cl(h) for some α- -open set U. Since H is strong β- -open, by Theorem1.15, s* cl (H) is semi - regular. Hence, H is an α - set. Example (1) Let X, τ and be as in Example 2.3. Then {a, c} is an α -set but not a semi- -open. (2) Let X = {a, b, c, d, e}, τ = {, X, {a}, {e}, {a, e}, {c, d}, {a, c, d}, {c, d, e},{b, c, d,e}, {a, c,d, e}} and ={ } then {b, c,d} is a semi- -open but not an α -set. Example (1) Let X, τ and be as in Example 2.3. Then {a, c} is an α - set but not a strong β- open. (2) Let X, τ and be as in Example 2.6. Then {a} is a strong β- -open but not an α - set. Theorem The following are equivalent for a subset H of an ideal topological space (X, τ, ):

5 (1) H is an α- -open. (2) H is a pre- -open and an α -set. Proof.(1) (2): Since every α- -open set is a pre- -open and an α -set, it is obvious. (2) (1): Let H be a pre- -open and an α -set. By Proposition 2.13, H is semi- open. Since H is semi- -open and pre- - open, by Proposition 1.6, it is an α- -open. Example (1) Let X = {a, b, c}, τ = {, {a}, {b}, {a, b}, X} and ={,{a}}.then {b, c} is an α - set but not a pre- -open. (2) In Example 2.12,{a, c} is a pre- -open but not an α - set. Theorem The following are equivalent for a subset H of an ideal topological space (X, τ, ): (1) H is an α -set. (2) H=A B where A is an -set and B is an - set. Proof.( ) : Let H be an α -set. This implies H = C D where C is α- -open and D is semi- -regular. By Lemma 1.18, we have C = E F where E is open and F is an -set. Moreover, we have H=C D=E F D=(E D) F such that A= E D is an -set and B=F is an -set. ( ) : Let H=A B where A is an - set and B is an -set. Since A is an -set, there exist an open set U and a semi- -regular set V such that A = U V. We have H=A B=U V B=(U B) V where U B is, by Lemma1.18,an α- - open. Thus, H is an α -set. Example (1) Let X = {a, b, c, d}, τ = {, {c}, {a, c}, {b, c}, {a, b, c},{a, c, d},x} and ={,{c}}.then{c} is an set but not an -set. (2 )Let X, τ and be as in Example 2.9. Then {a, b, c} is an -set but not an -set. Theorem Let (X, τ, ) be an ideal topological space. Then the following are equivalent. (1) X is - hyper connected. (2) Every semi- -open set is an -dense set. (3) Every α -set is an -dense set. (4) Every -set is an -dense set. Proof.(1) (2): Let H be a semi- -open set. Then there exists an open set G such that G H cl* (G). Since G is -dense, G* = X and so H*= X which implies that H is an -dense. (2) (3): is clear, by Proposition (3) (4): is clear, by Remark 2.2. (4) (1):Let H be a non empty open set. Then H is an -set and so by (4),H is an - dense set. Theorem Let H be a subset of an - submaimal ideal topological space (X, τ, )Then the following are equivalent. (1) H is at- -set. (2) H is a semi*- -closed set. (3) H is both an α*- -set and an α -set. Proof. (1) (2): H is a t- -set implies that int(h) = int(cl*(h)) which implies int(cl*(h)) H and so H is a semi*- -closed set. Conversely, if H is semi*- -closed, then int(cl*(h)) H and so it follows that int(cl*(h))=int(h).hence H is a t- -set. (2) (3): Clearly int(h) int(cl*(int(h))). But int(cl*(int(h))) int(cl*(h)) H and so int(cl*(int(h))) int(h). Hence int(cl*(int(h))) = int(h) which implies that H is an α*- -set. Also, H = H X where H is a t- -set by (1) and X is α- -open. Thus H is an α -set. (3) (1): Suppose H is both an α*- -set and an α -set. Th en H = U V where U is an α- - open and V is a t- -set. Now int (cl*(h)) = int (cl* (U V)) int [cl*(u) cl* (V)]=int

6 (cl*(u)) int (cl*(v)) = int (cl*(u)) int (V)=int [cl* (U) int (V)] int (cl* [U int (V)]) = int (cl* (int (U V))) = int (cl* (int (H))) = int (H) and so int (cl*(h)) int (H). But int (H) int(cl*(h)) which implies that H is a t- -set. Example Let X = {a, b, c, d}, τ = {, {a, b}, {a, b, c}, {a, b, d}, X} and = {, {b}}. Then {a} is an α*- -set but not an α - set and {a, b} is an α -set but not an α*- -set. This shows that α*- -sets and α - sets are independent of each other, in general. Theorem Let H be a subset of an -sub maximal ideal topological space(x, τ, )Then the following are equivalent. (1) H is semi- -regular set. (2) H is semi*- -closed set and an α -set. (3) H is an α*- -set and α -set. Proof.(1) (2): is clear. [see Theorem 1.21] (2) (3):is clear.[see Theorem 2.22] (3) (1): If H is an α -set,bytheorem 2.14, H is both a semi- -open and an α - set. Again, if H is an α*- -set, by Theorem 2.22, H is a t- -set and so H is a semi- - regular set. Remark The following examples show that the concepts of semi*- -closed set and α -set are independent of each other in general. Example Let X, τ and be as in Example 2.12.Then {c}is a semi*-i-closed but not an α -set. Example Let (R, τ) be the real numbers with the usual topology τ with ideal ={ }. Then R\{0} is an α -set, but it is not a semi*- -closed set. If ={ }, then H* =cl(h) and cl* (H)=cl(H) for every subset H of an ideal topological space. Let A = R \{0}. Then cl*(a) = R and int(cl* (A)) = R. Since int(cl* (A)) A, A is not a semi*- -closed set. On the other hand, A=A R where A is open (and hence α- -open) and R is semi- -regular. This shows that A is an α -set. Example Let X, τ and be as in Example2.12.Then {a, b } is an α -set but not an α* - -set and {c} is an α* - - set but not an α -set. This shows that α - sets and α*- -sets are independent of each other, in general. Theorem Let(X, τ, )bean* - extremely disconnected ideal space. Then the following property holds: α O(X)=α (X) where α O(X)denotes the family of α- -open subsets of X. Proof. We know that every α- -open set is α -set. Hence α O(X) α (X). Suppose H α (X). Then H = U V where U is an α- -open and V is a semi- -regular. Now V is semi- -regular implies that V is a t- -set and also a semi- -open. Hence int(v) = int(cl*(v)) and V cl* (int(v)) which implies that int(v) = int(cl* (V)) and cl* (V) = cl* (int(v)). Since X is * - extremally disconnected, int(cl* (int(v))) = cl* (int(v)) = cl* (V). Thus, int(v) = int(cl* (V)) = int(cl* (int(v)))= cl* (V) V and so V is open. We have U is α- -open set and V is open set. By Proposition 1.22, H = U V is α- -open. Therefore α (X) α O(X). Hence α O(X) =α (X). 3. Further Properties Theorem 3.1. The following are equivalent for a subset H of an ideal topological space (X, τ, ): (1) H is open, (2) H is α- -open and -set, (3) H is α- -open and weak -set, (4) H is α- -open and a -set.

7 Proof.(1) (2):Obvious. (2) (3): Obvious, by Remark1.10. (3) (4): Obvious, by Remark1.10. (4) (1): By the α- -openness of H, H int(cl* (int(h))) = int(cl* [int(u V)]) where U τ and V is an α*- -set. Hence H U H U int(cl* [int(u V)]) = U int (cl* [int(u) int (V)] U int[cl* (int(u)) cl* (int(v))] =U int(cl* (int(u))) int(cl* (int(v))) =U int(v) = int (U V) = int (H). This shows that H is open. Example 3.2. The notions of α- -open sets and -sets are independent as is shown in[23]. Example 3.3. See Remark Example 3.4. The notions of α- -open sets and -sets are independent as is shown in [16]. Definition 3.5. A subset H of an ideal topological space(x, τ, ) is called a -set if H = G B, where G is open and B is β- -closed. The family of all -sets of a space(x, τ, )will be denoted by (X). Theorem 3.6. Let H be a subset of an ideal topological space(x, τ, ).Then H (X) if and only if H=G β- -cl(h)for some open set G. Proof.( ) : Assume that H = G β- -cl(h) for some open set G. Since β- -cl(h) is β- -closed, H (X), ( ) : Let H (X). We have H=G A where G is open and A is β- - closed. Since H A, β- -cl(h) β- -cl(a) = A. Hence, G β- -cl(h) G A = H G β- -cl(h) and hence H = G β- -cl(h). Definition 3.7. A subset H of an ideal topological space (X, τ, ) is said to be (1) gβ- -closed if β- -cl(h) M whenever H M and M is open set in X. (2) gβ- -open if XH is gβ- -closed. Theorem 3.8. For a subset A of an ideal topological space (X, τ, ),the following are equivalent. (1) A is β- -closed, (2) A is a -set and gβ- -closed. Proof.(1) (2):Since every β- -closed set is a -set and gβ- -closed, it is completed. (2) (1):Let A be a -set. Then we have A=G β- -cl(a) for an open set G in X. We have A G. Since A is gβ- -closed, then β- -cl(a) G. Thus, β- -cl(a) G β- -cl(a) = A and also, A is β- -closed. Theorem 3.9.Let H be a subset of an ideal topological space (X, τ, ). If H (X),then (1) β- -cl(h) H is β- -closed. (2) H (X β- -cl(h)) is β- -open. Proof.(1)Let H (X).By Theorem 3.6, H=V β- -cl(h) for some open set V. Hence β- cl(h)\h = β- -cl(h)\(v β- -cl(h)) = β- -cl(h) (X\(V β- -cl(h))) = β- -cl(h) ((X\V) (X\β- -cl(h))) = (β- -cl (H) (X\V)) (β- -cl(h) (X\β- -cl(h))) = (β- -cl (H) (X\V)) = β- -cl(h) (X\V). Thus, β- -cl(h)\h is β- -closed. (2)Sinceβ- -cl(h)\hisβ- -closed,x\(β- -cl(h)\h)isβ- -open.hencex\(β- - cl(h)\h)=x\ (β- -cl(h) (X\H))=(X\β- -cl(h)) H.Thus,H (X\β- -cl(h))is β- -open. Remark We obtain the following diagram for the subsets stated above: α (X) (X) (X)W (X)

8 W LC(X) (X) (X) Remark In the above diagram, none of the implications is true as is shown by the following four Examples and the above Examples(2.5,2.6,2.7and2.8). (1) InExample2.23,{a}is weak -set but not an -set. (2) InExample2.23,{b}is -set but not weak -set. (3) InExample2.23,{b}is weakly- -locally closed set but not -set. (4) InExample2.20,{a}is -set but not weakly- -locally closed set. REFERENCES 1. A.Acikgoz, S.Yuksel and T.Noiri, α- -preirresolute functions and β- -preirresolute functions, Bull. Malays. Math. Sci. Sco., 28(2005), A. Acikgoz, T. Noiri and S. Yuksel, Onα- -continuous and α- -open functions, Acta Math. Hungar., 105 (1-2) (2004), A.Acikgoz and S.Yuksel, Some new sets and decompositions of A -R-continuity, α- - continuity, continuity via idealization, Acta Math. Hungar., 114(1-2)(2007), A. Acikgoz and S. Yuksel, Decompositions of some forms of continuity, Commun. Fac. Sci. Univ. Ank. Series A1, 56(1)(2007), M. Akdag, On b- -open sets and b- -continuous functions, Internat. J. Math. Math. Sci., (2007), J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology and its Applications, 93(1999), J. Dontchev, Between A- and B-sets, Math. Balkanica (N. S), 12(3-4)(1998), J.Dontchev, On pre- -open sets and a decomposition of -continuity, Banyan Math J., 2(1996). 9. Ekici E, On * sets, -sets * -sets and decompositions of continuity in ideal toplogical spaces, Analele Stiintifice Ale Universitatii Ii Al.I. Cuza Din Iasi (S.N.) Matematica, LIX(2013), E. Ekici, On pre- -open sets, semi- -open sets and b- -open sets in ideal topological spaces, Acta Universitatis Apulensis, 30(2012), E. Ekici and T. Noiri, Decompositions of continuity, α-continity and -continuity, Chaos, Solitons and Fractals, 41(2009), E. Ekici and T. Noiri, *-extremally disconnected ideal topological spaces, Acta Math. Hungar., 122(1-2)(2009), E. Ekici and T. Noiri, Properties of -submaimal ideal topological spaces, Filomat, 24(2010), N.Gowrisankar, A.Keskinand N.Rajesh, Quasi β- -continuous function Journal of Advanced Research in Pure Mathematics, 3(1)(2011), A.Gulhan and Y. Melike, On decompositions of some weak forms of continuity via idealization, Journal of Advanced Research in Pure Mathematics, 3(3)(2011), E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96(4)(2002), E. Hatir and T. Noiri, On β- -open sets and a decomposition of almost- -continuity, Bull. Malays. Math. Sci. Soc., (2)29(2)(2006), E. Hatir, A. Keskin and T. Noiri, On a new decomposition of continuity via idealization, JP Jour. Geometry and Topology, 3(1)(2003), E.Hatir, A.Keskinand T.Noiri, Anoteonstrong β- -sets and strong β- -continuous functions, Acta Math. Hungar., 108(2005), E. Hayashi, Topologies defined by local properties, Math. Ann., 156(1964), D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97(4)(1990), A.Keskin, T. Noiri and S. Yuksel, Idealization of a decomposition theorem, Acta Math. Hungar., 102(4)(2004), A. Keskin and S.Yuksel, Onsemi- -regular sets, I-sets and decompositions of continuity, R 1 C-continuity, continuity, Acta Math. Hungar., 113(3)(2006), A. Keskin, S. Yuksel and T. Noiri, Decompositions of -continuity and continuity, Commun. Fac. Sci. Univ. Ank. Series A1, 53(2004), K. Kuratowski, Topology, Vol. I, Academic Press (New York, 1966). 26. Q. M. Latif, Characterizations of α- -open sets and α- -continuity, King Fahd Univ., TR353, (2006), M. Rajamani, V. Inthumathi and S. Krishnaprakash, On decompositions of α- -continuity, Journal of Advanced Research in Pure Mathematics,4(2)(2012), V.Renukadevi, On subsets of ideal topological spaces, CUBO journal, 12(2)(2010), V.Renukadevi, Semiregular sets and -sets in ideal topological spaces, Journal of Advanced Research in Pure Mathematics, 2(3)(2010), V. Renukadevi, Strong β- -open sets, Journal of Advanced Research in Pure Mathematics, 2(2)(2010), V.Renukadevi, Note on R closed and - sets, Acta Math. Hungar., 122(4)(2009), [32] 32. J. Tong, A decomposition of continuity, Acta Math. Hungar., 48 (1-2) (1986), J. Tong, On decomposition of continuity in topological spaces, Acta Math. Hungar., 4(1-2)(1989), R.Vaidyanathaswamy, Set Topology, Chelsea Publishing Company (1946). 35. K. Viswanathan and J. Jayasutha, Some decompositions of continuity in ideal topological spaces, Eur. J. Math. Sci., 1(1)(2012),