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K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945 Research Paper α-SETS IN IDEAL TOPOLOGICAL SPACES 1K. V. Tamil Selvi, 2P. Thangaraj, 3O. 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 αsets 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, αcontinuity, 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))), 02010 Mathematics Subject Classification: 54A10, 54C10, 54D10, 54D15 11. β--closed[16] if its complement is β--open Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019 K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945 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 αI 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∈WLC(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 1.10. [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 1.11. [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 1.12. [22] Every regular--closed set is * -closed. Remark 1.13. [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 1.14. [29] Let (X, τ, ) be an ideal topological space and H ⊆ X. Then the following hold: s *cl(H) =H ∪ int (cl* (H)). Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019 K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945 Theorem 1.15. [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 1.17. [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 1.18. [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 1.19. [6] A subset H of an ideal topological space (X, τ, ) is said to be-dense if H* = X. Definition 1.20. [6] An ideal topological space (X, τ, ) is said to be -hyper connected if every non empty open subset of X is -dense. Theorem 1.21. [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 1.22. 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 1.23. [12] An ideal topological space X is called *-extremally disconnected if the *-closure of every open subset of X is open. Lemma 1.24. [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 Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019 K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945 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 2.10. In Example 2.6, {b} is an α- set but not an α- set. Example 2.11. Let X, τ and be as in Example2.5.Then {a, b, d } is an α - set but not a - set. Example 2.12. Let X={a, b, c},τ={∅,{a, b}, X} and ={∅,{c}}. Then {b, c} is an α- set but not an α-set. Proposition 2.13. 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 2.15. (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 2.16. (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 2.17. The following are equivalent for a subset H of an ideal topological space (X, τ, ): Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019 K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945 (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 2.18. (1) Let X = {a, b, c}, τ = {∅, {a}, {b}, {a, b}, X} and ={∅,{a}}.Then {b, c} is an αset but not a pre--open. (2) In Example 2.12,{a, c} is a pre- -open but not an α- set. Theorem 2.19. 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 2.20. (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 anset 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 2.21. 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 2.13. (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 2.22. 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 Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019 K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945 (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 2.23. 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 2.24. 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 αset. 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 2.25. The following examples show that the concepts of semi*--closed set and α-set are independent of each other in general. Example 2.26. Let X, τ and be as in Example 2.12.Then {c}is a semi*-I-closed but not an α-set. Example 2.27. 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 2.28. 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 2.29. 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) ⊆ αO(X). Hence αO(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. Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019 K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945 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 1.10. 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 3.10. We obtain the following diagram for the subsets stated above: α(X) (X) (X)W (X) Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019 K. V. 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