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International Journal of Mathematical Archive-5(2), 2014, 24-29 Available online through www.ijma.info ISSN 2229 – 5046 A NOTE ON d-LOCALLY CLOSED SETS H. Jude Immaculate*1 & I. Arockiarani2 1,2Department of Mathematics, Nirmala College for Women, Coimbatore, (T.N.), India. (Received on: 23-01-14; Revised & Accepted on: 12-02-14) ABSTRACT In this paper we devise a new class of generalized open sets in a topological space called d-open sets. Using this new class of sets we deduce d-t sets, d-B-sets and locally d-closed sets and some of their properties and related concepts are investigated. Keywords: d-open, d-t-set, d-B-set, d-preopen, d-semiopen, locally d-closed. 1. INTRODUCTION In recent years a number of generalizations of open sets have been developed by many Mathematicians. N. Levine[21] and M.E Abd El-Monsef [1] introduced semiopen sets and β-sets respectively. In [21]Levine defined a subset A of a topological space (X, τ) to be semiopen if there exists an open set U in X such that U⊂A⊂𝑈𝑈. β-sets are also called as semi-preopen sets by Andrijevic [3]. In [15] O. Njastad defined a subset A of a topological space (X, τ) to be a α-set if A⊂int(cl(int(A)). Andrijevic[6] introduced a class of generalized open sets in a topological space, so called b-open sets [2]. Tong [16] introduced the concept of t-set and B-set in topological spaces. In this paper, we introduce the notion of d-open sets, locally d-closed sets, d-t-set and d-B-set. Throughout this paper (X,τ) and (Y,σ) stands for topological spaces with no separation axioms assumed, unless otherwise stated. The complement of a d-open set is said to be dclosed. The family of all d-open (resp. α-open, semi open, pre open, regular open) subsets of a space X is denoted by DO(X)(resp. αO(X), SO(X), PO(X), RO(X))respectively. 2. PRELIMINARIES We present here relevant preliminaries required for the progress of this paper Definition: 2.1 A subset A of a space X is said to be 1. Preopen [12] if A⊂int(cl(A)) 2. Semiopen [11] if A⊂ cl(int(A)) 3. α-open[2] if A⊂ int(cl(int(A))) 4. β-open or semi-preopen [3] if A⊂cl(int(cl(A))) 5. Regular open[7] if A= int(cl(A)) Definition: 2.2: A subset A of a space X is called 1. t-set if [16]int(A) = int(cl(A)) 2. B-set if [16] A= U∩V, where U∈τ and V is a t-set. 3. Locally closed [7] if A=U∩V where U∈τ and V is a closed set. Proposition: 2.3 [3] Let A be a subset of a space X. Then 1. scl(A) = A∪int(cl(A)), sint(A) = A∩cl(int(A)). 2. pcl(A) = A∪cl(int(A)), pint(A) = A∩int(cl(A)). 3. spcl(A) = A∪int(cl(int(A))), spint(A) = A∩cl(int(cl(A))). 4. 𝑐𝑐𝑐𝑐α (A) = A∪cl(int(cl(A))), 𝑖𝑖𝑖𝑖𝑖𝑖α (A) = A∩int(cl(int(A))) 1,2Department Corresponding author: H. Jude Immaculate*1 of Mathematics, Nirmala College for Women, Coimbatore, (T.N.), India. E-mail: [email protected] International Journal of Mathematical Archive- 5(2), Feb. – 2014 24 1 2 H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014. Proposition: 2.4 [2] Let A be a subset of a space X. Then 1. sint(cl(A))=cl(int(cl(A))) 2. scl(int(A))=int(cl(int(A))) 3. 𝑐𝑐𝑐𝑐α (int(A))=cl(int(A))=cl(𝑖𝑖𝑖𝑖𝑖𝑖α (A))= 𝑐𝑐𝑐𝑐α (𝑖𝑖𝑖𝑖𝑖𝑖α (A)) 4. int(𝑐𝑐𝑐𝑐α (A))=int (cl(A))= 𝑖𝑖𝑖𝑖𝑖𝑖α(cl(A)) 5. spcl(int(A))=int(cl(int(A)))=int(spcl(A)) 6. cl(spint(A))=cl(int(cl(A)) 3. d-OPEN SETS Definition: 3.1 A subset A of a space X is called d-open if A⊂scl(int(A)) ∪ sint(cl(A)). The class of all d-open sets in X is denoted by DO(X). The complement of a d-open set is a d-closed set. Proposition: 3.2 1. Every open set is d-open set. 2. Every preopen set is d-open set. 3. Every α-open set is d-open set. 4. Every semiopen set is d-open set. 5. Every regular open set is d-open set. Remark: 3.3 The converse of the above results need not be true may be seen by the following example. Example: 3.4 Let X= {a, b, c, d}, τ= {X,φ, {a}, {b},{c},{a, b},{a, c},{b, c},{a, b, c}} A= {a, d} is d-open but not open, preopen ,α-open and regular open. Example: 3.5 Let X= {a, b, c}, τ= {X,φ, {a}, {b, c}}. A= {b} is d-open but not semiopen. The following diagram holds for a subset A of a space X. 1. Open set 2. Regular open set 3. α-open set 4. d-open set 5. Semiopen set 6. Preopen set Theorem: 3.6 Let X be a topological space and A⊂X. Then A is d-open if and only if A=𝑖𝑖𝑖𝑖𝑖𝑖α A ∪ spintA. Proof: Let A be d-open. Then A⊂scl(int(A))∪sint(cl(A)) By proposition 2.3 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA= (A∩int(cl(int(A))))∪(A∩cl(int(cl(A)))) = A ∩(int(cl(int(A)))∪cl(int(cl(A)))) = A ∩(scl(int(A))∪sint(cl(A))) = A Conversely suppose A= 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA = (A∩int(cl(int(A))))∪(A∩cl(int(cl(A)))) = (A∩scl(int(A))∪(A∩sint(cl(A)) ⊂ scl(int(A))∪sint(cl(A)) Hence A is d-open. © 2014, IJMA. All Rights Reserved 25 1 2 H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014. Theorem: 3.7 1. The union of any family of d-open set is d-open. 2. The intersection of a open set and a d-open set is a d-open set Remark: 3.8 Every d-open set can be represented as a union of β-open set and a α-open set. Proposition: 3.9 Let A be a d-open set such that int(A) = φ.Then A is a β-open set. Definition: 3.10 A subset A of a space X is called d-closed set if X\A is d-open. Thus A is d-closed if and only if sint(cl(A))∩scl(int(A))⊂A. Definition: 3.11 If A is a subset of a space X then d-closure of A is denoted by dcl(A) is the smallest d-closed set containing A. The d-interior of A is denoted by dint (A) is the largest d-open set contained in A. Theorem: 3.12 The following holds for a subset A of a space X dcl(A)=𝑐𝑐𝑐𝑐α (A)∩spcl(A). Proof: It is clear that dcl(A)⊂ 𝑐𝑐𝑐𝑐α (A)∩spcl(A). Conversely, Since dcl(A) is a d-closed set we have dcl(A) ⊃ sint(cl(dcl(A)))∩scl(int(dcl(A))) ⊃ sint(cl(A))∩scl(int(A)) Hence by proposition 2.3 𝑐𝑐𝑐𝑐α (A)∩spcl(A) = (A∪cl(int(cl(A))))∩(A∪int(cl(int(A)))) = (A∪sint(cl(A)))∩( A∪ (scl(int(A))) ⊂ dcl(A) Definition: 3.13 Let A be a subset of a space X. The d-interior of A is denoted by dint(A) is defined by the union of all d-open sets contained in A. Theorem: 3.14 The following holds for a subset A of a space X dint(A)= 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA. Proof: It is clear that dint(A) ⊃ 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA. Conversely dint(A) ⊂ scl(int(dint(A)))∪sint(cl(dint(A))) ⊂scl(int(A))∪sint(cl(A)) By proposition 2.3 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA= A ∩(scl(int(A)))∪(A ∩sint(cl(A))) ⊃ dint(A) Proposition: 3.15 1. dcl(int(A)) = 𝑐𝑐𝑐𝑐α (int(A))∩spcl(int(A)). = cl(int(A))∩int(cl(int(A))) = int(cl(int(A))) 2. int(dcl(A)) = int(𝑐𝑐𝑐𝑐α (A))∩int(spcl(A)) = int(cl(A))∩int(cl(int(A))) = int(cl(int(A))) 3. 𝑖𝑖𝑖𝑖𝑖𝑖α (dcl(A))= 𝑖𝑖𝑖𝑖𝑖𝑖α (𝑐𝑐𝑐𝑐α (A))∩ 𝑖𝑖𝑖𝑖𝑖𝑖α (spcl(A)) = int(cl(A))∩int(cl(int(A))) = int(cl(int(A))) From the above we conclude that dcl(int(A))= int(dcl(A))= 𝑖𝑖𝑖𝑖𝑖𝑖α (dcl(A))= int(cl(int(A))) Proposition: 3.16 1. cl(dint(A)) = cl 𝑖𝑖𝑖𝑖𝑖𝑖α(A))∪cl(spint(A)) = cl(int(A))∪cl(int(cl(A))) = cl(int(cl(A))) 2. dint(cl(A)) = 𝑖𝑖𝑖𝑖𝑖𝑖α(cl(A))∪spint(cl(A)) = int(cl(A))∪cl(int(cl(A))) = cl(int(cl(A))) © 2014, IJMA. All Rights Reserved 26 1 2 H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014. From the above we conclude that cl(dint(A))= dint(cl(A)) = cl(int(cl(A))) 4. d-t-sets Definition: 4.1 A subset A of a space X is said to be; 1. d-t-set if int(A)=int(dcl(A)); 2. d-B-set if A=U∩V, where U∈τ and V is d-t-set; 3. d-semiopen if A⊆cl(dint(A)); 4. d-preopen if A⊆int(dcl(A)). Proposition: 4.2 For subsets A and B of a space (X, τ), the following properties hold; 1. A is a d-t-set if and only if it is d-semi closed. 2. If A is d-closed, then it is a d-t-set. 3. If A and B are d-t-sets, then A∩B is a d-t-set. Proof: 1. Let A be a d-t-set. Then int(A)=int(dcl(A)). Therefore int(dcl(A)) ⊆int(A) ⊆A and A is d-semi closed. Conversely if A is d-semi closed, then int(dcl(A) ⊆A thus int(dcl(A)) ⊆int(A). Also A⊆dcl(A) and int(A)⊆ int(dcl(A)). Hence int(A)=int(dcl(A)). 2. Let A be d-closed, then A=dcl(A) and int(A)=int(dcl(A)) therefore A is d-t-set. 3. Let A and B be a d-t-set. Then we have int(A∩B) ⊆ int(dcl(A∩B)) ⊆(int(dcl(A))∩(dcl(B))) =int(dcl(A)∩Int(dcl(B)) =int(A)∩Int(B) =int(A∩B) Then int(A∩B)=int(dcl(A∩B) hence A∩B is a d-t-set. Proposition: 4.3 For a subset A of a space (X, τ), the following properties hold; i. If A is t-set then it is a d-t-set; ii. If A is d-t-set then it is a d-B-set; iii. If A is B-set then it is a d-B-set. Theorem: 4.4 For a subset A of a space (X, τ),the following are equivalent; 1. A is open. 2. A is d-preopen and a d-B-set. Proof: (1)⇒(2) Let A be open. Then A ⊆dcl(A), A=Int(A) ⊆ Int(dcl(A)) and A is d-preopen. Also A= A∩X hence A is d-B-set. (2)⇒(1) Since A is a d-B-set, we have A=U∩V, where U is open set and Int(V)=Int(dcl(V)).By hypothesis, A is also d-preopen, and we have A ⊆ int(dcl(A) = int(dcl(U∩V) ⊆ int(dcl(U)∩dcl(V)) = int(dcl(U)∩int(dcl(V)) = int(dcl(U)∩Int(V) Hence A = U∩V= (U∩V)∩U ⊆ (int(dcl(U))∩int(V))∩U = (int(dcl(U))∩U)∩int(V) = U∩int(V) Therefore A= (U∩V) = (U∩int(V)), and A is open. Remark: 4.5 d-preopen sets and d-B-open sets are independent. Example: 4.6 Let X= {a, b, c} and τ={X,φ, {c}, {d}, {c, d}, {a, c, d}, {b, c, d}}. It is clear that {b, c} is a d-B-set but it is not d-preopen, since {b, c} = {b, c, d}∩{b, c}. {b, c} is b-t-set and {b, c}⊄Int(dcl({b, c})))={c}. © 2014, IJMA. All Rights Reserved 27 1 2 H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014. Example: 4.7 Let X= {a, b, c} and τ={X,φ, {b}, {a, b}, {b, c, d}}.It is clear that {b, c} is a d-preopen but it is not a dB-set, since {b, c} ⊆ Int(dcl({b, c}))=X, since {c, b} is not d-t-set and the open set containing {b, c} is X or {b, c, d}, therefore {b, c} is not d-B-set. Lemma: 4.8 Let A be an open subset of a space X. Then dcl(A)= int(cl(int(A)) and int(dcl(A))=int(cl(int(A)) Proof: Let A be open set, then dcl(A) = 𝑐𝑐𝑐𝑐α (A)∩spcl(A). = A∪(int(cl(int(A))))∩cl(int(cl(A)))) = A∪( int(cl(int(A))))∩cl(A)) = A∪( int(cl(int(A)))) = Int(cl(int(A))) 5. LOCALLY D-CLOSED SETS Definition: 5.1 A subset A of a space X is called locally d-closed if A=U∩V where U∈τ and V is a d-closed set. Theorem: 5.2 Let A be a subset of X, A is locally d-closed if and only if there exists an open set U⊆X such that A=U∩ dcl(A) Proof: Since A is locally d-closed, then A=U∩F where U is open and F is d-closed. So A⊆ U and A ⊆F then A⊆ dcl(A) ⊆dcl(F)= F. ∴A⊆U∩dcl(A)⊆U∩dcl(F)=U∩F=A. ∴ A=U∩ dcl(A) Conversely, assume that A=U∩ dcl(A). Since dcl(A) is d-closed A is locally d-closed Result: 5.3 1. Every closed set is locally d-closed. 2. Every locally closed is locally d-closed. Example: 5.4 Let X= {a, b, c, d} and τ= {X,φ, {a},{c},{a, c},{b, c},{a, b, c}}. A= {b} is locally d-closed but not locally closed. And A= {a} is locally d-closed but not closed. Proposition: 5.5 Let A be a subset of a topological space X if A is locally d-closed then i. dcl(A)-A is a d-closed set. ii. [A∪(X-dcl(A))] is d-open. iii. A ⊆ dint[A∪(X-dcl(A))]. Proof: i. Let A be a subset of a topological space X and A be locally d-closed then there exists an open set U such that A=U∩ dcl(A) dcl(A)-A= dcl(A)-(U∩dcl(A)) = dcl(A)∩(U∩dcl(A)) 𝑐𝑐 = dcl(A)∩[X-(U∩dcl(A))] = dcl(A)∩[(X-U)∪(X-dcl(A)] = [dcl(A)∩(X-U)]∪φ = dcl(A) ∩(X-U) Therefore dcl(A)-A is d-closed ii. Since dcl(A)-A is d-closed then[X-(dcl(A)-A)] is d-open [X-(dcl(A)-A)] = X-(dcl(A) ∩𝐴𝐴𝑐𝑐 ) = (X-dcl(A))∪(X-𝐴𝐴𝑐𝑐 ) = (X-dcl(A)) ∪A ⇒ [A∪(X-dcl(A))] is d-open. iii. It is clear that A⊆ [A∪(X-dcl(A))] = dint[A∪(X-dcl(A))] Hence A ⊆ dint[A∪(X-dcl(A))] © 2014, IJMA. All Rights Reserved 28 1 2 H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014. Definition: 5.6 A subset A of a topological space is called D(c, d) set if int(A) = dint(A). From the following examples we can deduce that d-open and D(c, d) sets are independent. Example: 5.7 Let X= {a, b, c, d}, τ= {X,φ, {a}, {b},{c},{a, b},{a, c},{b, c},{a, b, c}} DO(X)= {X,φ, {a},{b},{c},{a, b},{a, c}, {a, d},{b, c},{b, d}, {c, d},{a, b, c},{a, b, d},{a, c, d},{b, c, d}}. It is clear that A= {d} is D(c, d) set but not d-open .Also B= {a, b, d} is d-open but not D(c, d) Theorem: 5.8 For a subset A of a space X the following are equivalent. 1. A is open 2. A is d-open and D(c, d) set. Proof: 1⇒ 2 If A is open then A is d-open and A=int(A)=dint(A) So A is a D(c, d) set. 2⇒1 A is d-open and D(c, d) set. Then A= dint (A) and int(A)=dint(A) and consequently A is open. Proposition: 5.9 Suppose X is closed under finite union of d-closed sets. Let A and B be locally d-closed. If A and B are separate then A∪B is locally d-closed. Proof: Since a and B are locally d-closed A=G ∩ dcl(A) and B=H ∩ dcl(B) where g and H are open in X. Put U= G ∩(X\cl(B)) and V=H ∩ (X\cl(A)). Then U ∩ dcl(A) = G ∩(X\cl(B)) ∩ dcl(A) = A ∩(X\cl(B)) = A Since A⊆ X\cl(B), Similarly V∩dcl(B )=B.and U∩dcl(B)⊆ U∩cl(B)=φ and V ∩ dcl(A)=V ∩cl(A) = φ Since U and V are open (U ∩ V) ∩ dcl(A∪B) = (U∩V)∩(dcl(A)∪dcl(B)) = (U∩dcl(A))∪(U∩dcl(B))∪(V∩dcl(A))∪(V∩dc l(B)) = A∪ B Hence A ∪ B is locally d-closed. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. M.E. Abd El-Monsef, S.N.El-Deeb and R.A Mahmoud, “β -open sets and β -continuous mappings”, Bull. Fac. Sci. Assiut Univ. 12(1983), 77-90 D. Andrijevic, “Some properties of the topology of α-sets”, Mat. Vesnik 36(1984). D. Andrijevic, “Semi-preopen sets”, Mat. Vesnik 38(1) (1986), 24-32. D. Andrijevic, “On the topology generated by preopen sets, Ibid. 39(1987), 367-376. D. Andrijevic, “On SPO-equivalent topologies. Suppl. Rend. Circ. Mat. Palermo 29(1992), 317-328 D. Andrijevic, “On b-open sets”, Mat. Vesnik 48(1996), 59-64. N. Bourbaki, “General Topology”, Part 1, Addison-Wesley (Reading Mass, 1966). Y. Erguang and T. Pengfei “On decomposition of A-Continuity” Acta Math. Hungar 110(4) (2006), 309-313. M.Ganster nd I.L. Reilly, “Locally closed sets and LC-continuous functions” International Journal of Mathematics and Mathematical sciences, Vol.12, (1989), 239-249. M.Ganster and D. Andrijevic, “On some questions concerning semi-preopen sets, Journ. Inst. Math. and Comp. Sci.(Math. Ser) 1(1988), 65-75 N.Levine, “Semi-open sets and semi-continuity in topological spaces”, Amer Math. Monthly 70(1963), 36-41. A.S. Mashhour, Abd El-Monsef M.E. and S.N.El-Deeb,” On pre-continuous and weak continuous mappings” S.N., Proc. Math Phys. Soc. Egypt, 53, (47-53) (1982) A.S. Mashhour, I.A. Hasanein and S.N.El-Deeb, “α-continuity and α-open mappings” Acta. Math. Hungar. 41 (1983), 213-218. A.A. Nasef, “On b-locally closed sets and related topics”, Chaos solutions & Fractals 12(2001).1909-1925. O. Njastad, “On some classes of nearly open sets”, Pacific. J. Math. 15(1965), 961-970. J. Tong, “On Decomposition of continuity in topological spaces” Acta Math. Hungar. 54 (1-2) (1989).51-55. Source of support: Nil, Conflict of interest: None Declared © 2014, IJMA. All Rights Reserved 29