Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Asian Journal of Current Engineering and Maths1: 4 Jul – Aug (2012) 173 – 176. Contents lists available at www.innovativejournal.in ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Journal homepage : http://www.innovativejournal.in/index.php/ajcem On Semi*- Closed Sets A. Robert1, S. Pious Missier2 1Department 2P.G. of Mathematics, Aditanar College, Tiruchendur, India-628216. Department of Mathematics, V.O.Chidambaram College, Thoothukudi, India-628008. ARTICLE INFO ABSTRACT Corresponding Author A. Robert Department of Mathematics, Aditanar College, Tiruchendur, India-628216. Abstract: In this paper we introduce a new class of sets, namely semi*-closed sets. We give characterizations for semi*-closed sets. We define the semi*closure, semi*-limit points and semi*-derived set of a subset. Further we investigate fundamental properties of semi*-closed sets. Key Words: Semi-Closed, SemiClosure, Semi-Limit Point, Semi*Open, Semi*-Closed, Semi*-Closure, Semi*-Limit Point. Mathematics Subject Classification: 54A05. ©2012, AJCEM, All Right Reserved. 1. INTRODUCTION N.Levine [7] introduced semi-open sets in topological spaces in 1963. N.Biswas [1] defined semi-closed sets. S.G.Crossley and S.K.Hildebrand [2] defined semi-closure of a set. N.Levine [8] defined and studied generalized closed sets in 1970. P.Das [3] defined semi-limit point of a subset. The semi-derived set of a subset of a topological space was also defined and studied by him. W.Dunham [5] introduced the concept of generalized closure and defined a new topology τ* and investigated some of their properties. G.Di Maio and T.Noiri [4] defined and studied semi-regular subsets. Quite recently the authors [11] introduced and studied semi*-open sets using the generalized closure operator. In this paper, we introduce the concept of semi*closed sets. We further show that the class of semi*-closed sets lies between the class of semi-closed sets and the class of closed sets. We give characterizations of semi*-closed sets. We investigate many fundamental properties of semi*-closed sets. We also define the semi*-closure of a subset and study their properties. 2. PRELIMINARIES Throughout this paper (X, τ) will always denote a topological space on which no separation axioms are assumed, unless explicitly stated. If A is a subset of the space (X, τ), Cl(A) and Int(A) denote the closure and the interior of A respectively. Also 𝓕 denotes the class of all closed sets in the space (X, τ). Definition 2.1: A subset A of a space X is (i) generalized closed (briefly g-closed) [8] if Cl(A)⊆U whenever A⊆U and U is open in X. (ii) generalized open (briefly g-open) [8] if X\A is g-closed in X. Definition 2.2: If A is a subset of X, (i) The generalized closure [5] of A is defined as the intersection of all g-closed sets in X containing A and is denoted by Cl*(A). (ii) the generalized interior of A is defined as the union of all g-open sets in X that are contained in A and is denoted by Int*(A). Definition 2.3: A subset A of a topological space (X, τ) is (i) semi-open [7] (resp. semi*-open [11] ) if there is an open set U in X such that U⊆A⊆Cl(U) (resp. U⊆A⊆Cl*(U)) or equivalently if A⊆Cl(Int(A)) (resp. A⊆Cl*(Int(A))). (ii) semi-closed [1](resp. pre-closed [9] and -closed [10]) if Int(Cl(A))⊆A ( resp. Cl(Int(A)⊆A and Cl(Int(Cl(A)))⊆A ). The class of all semi-open (resp. semi-closed, semi*-open) sets is denoted by SO(X, τ) (resp. SC(X, τ), S*O(X, τ)). Definition 2.4: [8] A topological space X is T1/2 if every gclosed set in X is closed. Definition 2.5: [5] If (X, τ) is a topological space, let τ* be the topology on X defined by the Kuratowski closure operator Cl*. That is, τ*={U⊆X : Cl*(X\U)=X\U}. Theorem 2.6: [5] If (X, τ) is a topological space, then (X, τ*) is T1/2. Definition 2.7: A space X is locally indiscrete [12] if every open set in X is closed. Definition 2.8: [6] The topology on the set of integers generated by the set S of all triplets of the form {2n-1, 2n, 2n+1} as sub base is called the Khalimsky topology and it is denoted by κ. The set S∪{2n+1:n∈ℤ} is a base for the topology κ. The digital line equipped with the Khalimsky topology is called the Khalimsky line. A subset A is open in (ℤ, κ) iff whenever x in A is even, x-1, x+1∈A. A subset A is closed in (ℤ, κ) iff whenever x in A is odd, x-1, x+1∈A. The topological product of two Khalimsky lines (ℤ, κ) is called the Khalimsky plane (ℤ2, κ2). Theorem 2.9: [11] every open set is semi*-open. Theorem 2.10:[11] Every semi*-open set is semi-open. 3. SEMI*-CLOSED SETS Definition 3.1: A subset A of a space (X, τ) is called a semi*-closed set if X\A is semi*-open. 173 Robert et.al/On Semi*- Closed Sets Notation: The class of all semi*-closed sets in (X, τ) is denoted by S*C(X, τ) or simply S*C(X). Definition 3.2: The semi*-closure of A is defined as the intersection of all semi*-closed sets in X containing A. It is denoted by s*Cl(A). Definition 3.3: Let A⊆X. An element x∈X is called a semi*adherent point of A if every semi*-open set in X containing x intersects A. Definition 3.4: Let A⊆X. An element x∈X is called a semi*limit point of A if every semi*-open set in X containing x intersects A in a point different from x. Definition 3.5: The set of all semi*-limit points of A is called the semi*-Derived set of A. It is denoted by Ds*[A]. Definition 3.6: A subset A of X is called semi*-regular if it is both semi*-open and semi*-closed. Theorem 3.7: A subset A of a space (X, τ) is semi*-closed if and only if there is a closed set F in (X, τ) such that Int*(F)⊆A⊆F. Proof: Necessity. Suppose A is semi*-closed. Then X\A is semi*-open. Then there exists an open set U in X such that U⊆X\A⊆Cl*(U) which implies X\U⊇A⊇X\Cl*(U). Note that in any space, X\Cl*(U)=Int*(X\U). Therefore X\U⊇A⊇Int*(X\U) where X\U is closed in X. Sufficiency. Suppose there is a closed set F in (X, τ) such that Int*(F)⊆A⊆F which implies X\Int*(F)⊇X\A⊇ X\F. Since X\Int*(F)=Cl*(X\F), we have Cl*(X\F)⊇X\A⊇X\F where X\F is an open set. Hence by Definition 2.3, X\A is semi*-open. Therefore A is semi*-closed. Remark 3.8: In any space (X, τ), ϕ and X are semi*-closed sets. Theorem 3.9: A⊆X is semi*-closed if and only if Int*(Cl(A))⊆A. Proof: Note that if A⊆B then X\A⊇X\B and Cl*(X\A)=X\Int*(A) and X\Cl*(A)=Int*(X\A). A is semi*-closed if and only if X\A is semi*-open. That is, if and only if X\A⊆Cl*(Int(X\A)) (By Definition2.3(i)). That is, if and only if Int*(Cl(A))⊆A. Theorem 3.10: If A is semi*-closed in X and B⊆X is such that Int*(A)⊆B⊆A. Then B is semi*-closed in X. Proof: Since A is semi*-closed, X\A is semi*-open. Now Int*(A)⊆B⊆A which implies X\Int*(A)⊇X\B⊇X\A. That is, Cl*(X\A)⊇X\B⊇X\A. Therefore by Theorem 3.39 [11], X\B is semi*-open. Hence B is semi*-closed. Theorem 3.11: If {Aα} is a collection of semi*-closed sets, then ⋂Aα is also a semi*-closed set. Proof: Since each Aα is semi*-closed, X\Aα is semi*-open. By Theorem 3.11[11], ∪(X\Aα) is semi*-open. That is, X\(⋂𝐴𝛼 ) is semi*-open. Hence ⋂Aα is semi*-closed. Corollary 3.12: If A is semi*-closed and U is semi*-open in X, then A\U is semi*-closed in X. Proof: Follows from A\U=A⋂(X\U). Remark 3.13: Union of two semi*-closed sets need not be semi*-closed as seen from the following example. Example 3.14: Let X={a, b, c, d} with τ={ϕ, {a}, {b}, {a, b}, {a, b, c}, X}. In the space (X, τ), the subsets A={a, c} and B={b ,c} are semi*-closed but A∪B={a, b, c}is not semi*-closed. Theorem 3.15: If A is semi*-closed in X and B is closed in X, then A∪B is semi*-closed in X. Proof: Since A is semi*-closed, X\A is semi*-open in X. Also X\B is open. By Theorem 3.10[11], (X\A)⋂(X\B)=X\(A∪B) is semi*-open in X. Hence A∪B is semi*-closed in X. Theorem 3.16: If A is any subset of a space X, then s*Cl(A) is semi*-closed. In fact s*Cl(A) is the smallest semi*-closed set in X containing A. Proof: Follows from Definition 3.1 and Theorem 3.11. Theorem 3.17: A is semi*-closed if and only if s*Cl(A)=A. Proof: A is semi*-closed implies s*Cl(A)=A is obvious. On the other hand, let s*Cl(A)=A. By Theorem 3.16, s*Cl(A) is semi*-closed and hence A is semi*-closed. Theorem 3.18: Every closed set is semi*-closed. Proof: Let A be closed. Then X\A is open. By Theorem 2.9, X\A is semi*-open. Hence A is semi*-closed. Remark 3.19: The converse of Theorem 3.18 is not true as shown by the following example. Example 3.20: In the space (X, τ) of Example 3.14, the subsets A={a, c} and B={b, c} are semi*-closed but not closed. Corollary 3.21: If A is semi*-closed and F is closed in X, then A⋂F is semi*-closed in X. Proof: Since F is closed, by Theorem 3.18, F is semi*closed. Then by Theorem 3.11, A⋂F is semi*-closed. Corollary 3.22: If A is semi*-closed and U is open in X, then A\U is semi*-closed in X. Proof: Follows from A\U=A⋂(X\U) and from Corollary 3.21. Theorem 3.23: Every semi*-closed set is semi-closed. Proof: If A is semi*-closed, X\A is semi*-open. By Theorem 2.10, X\A is semi-open. Hence A is semi-closed. Remark 3.24: The converse of Theorem 3.23 is not true as shown by the following example. Example3.25: Consider the space in Example 3.14. In the space (X, τ), the subsets {a, d} and {b, d} are semi-closed but not semi*-closed. Theorem 3.26: In any topological space (X, τ), 𝓕⊆S*C(X, τ)⊆SC(X, τ). Proof: Follows from Theorem 3.18 and Theorem 3.23. Remark 3.27: (i) If (X, τ) is a locally indiscrete space, then 𝓕=S*C(X, τ)=SC(X, τ). (ii) In the Sierpinski space (X, τ), where X={0, 1} and τ = {, {1}, X}, 𝓕=S*C(X, τ)=SC(X, τ). (iii) The inclusions in Theorem 3.26 may be strict and equality may also hold. This can be seen from the following examples. Example 3.28: Consider the space (X, τ) where X={a, b, c, d} and τ ={ϕ, {a} ,{ b, c, d}, X} 𝓕={ϕ, {a}, {b, c, d}, X}. Here 𝓕=S*C(X, τ)=SC(X, τ). Example 3.29: Consider the space (X, τ) where X={a, b, c, d} and τ ={ϕ, {a}, {b}, {a, b}, {b, c}, {a, b, c}, X}. Here 𝓕={ϕ, {d}, {a, d}, {c, d}, {a, c, d}, {b, c, d}, X}. SC(X, τ)=S*C(X, τ)={ϕ,{a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {c, d},{a, c, d}, {b, c, d}, X}. Here 𝓕 ⊊ S*C(X, τ)=SC(X, τ). Example 3.30: Consider the space (X, τ) where X={a, b, c, d} and τ={ϕ, {a, b}, X}. 𝓕={ϕ, {c, d}, X}; SC(X, τ) ={ϕ, {c}, {d}, {c, d}, X}; S*C(X, τ)={ϕ, {c, d}, X}. Here 𝓕=S*C(X, τ)⊊SC(X, τ). Example 3.31: Consider the space (X, τ) where X={a, b ,c, d} and τ ={ϕ, {a} ,{a, b, c}, X} 𝓕={ϕ, {d}, {b, c, d}, X}; SC(X, τ)={ϕ, {b}, {c}, {d}, {b, c}, {b, d}, {c, d}, {b, c, d}, X}; S*C(X, τ)={ϕ,{d}, {b, c}, {b, c, d}, X}. Here 𝓕⊊S*C(X, τ)⊊SC(X, τ). Remark 3.32: In a T1/2 space, the class of semi*-closed sets and the class of semi-closed sets coincide. In particular, in 174 Robert et.al/On Semi*- Closed Sets the Khalimsky line and in the real line with usual topology the semi*-closed sets and the semi-closed sets coincide. In any topological space, X\{x} is semi*-closed if and only if it is closed. Remark 3.33: The concepts of semi*-closed sets and αclosed sets are independent as seen from the following examples: Example 3.34: In the space (X, τ) where X={a, b, c, d} and τ={ϕ, {a}, {a, b, c}, X}, the subsets {b}, {c}, {b, d} and {c, d} are α-closed but not semi*-closed. Example 3.35: In the space (X, τ) where X={a, b, c, d} and τ={ϕ,{a},{b},{a, b}, {a, b, c}, X}, the subsets {a, c} and {b, c} are semi*-closed but not α-closed. Remark 3.36: The concepts of semi*-closed sets and preclosed sets are independent as seen from the following examples: Example 3.37: In the space (X, τ) where X={a, b, c, d} and τ={ϕ, {a}, X}, the subsets {b, d} and {c, d} are pre-closed but not semi*-closed. Example 3.38: In the space (X, τ) where X ={a, b, c, d}and τ={ϕ, {a}, {b}, {a, d}, {b, d}, {a, b, c}, X}, the subsets {a, c} and {b, c} are semi*-closed but not pre-closed. From the above discussions we have the following diagram: Diagram 3.39: Theorem 3.40: In any topological space (X, τ) ,the following results hold: (i) s*Cl(ϕ)=ϕ. (ii) s*Cl(X)=X. If A and B are subsets of X, (iii) A⊆s*Cl(A). (iv) A⊆B⟹s*Cl(A)⊆s*Cl(B). (v) s*Cl(s*Cl(A))=s*Cl(A). That is, the operator s*Cl is idempotent. (vi) A⊆sCl(A)⊆s*Cl(A)⊆Cl(A). (vii) s*Cl(A∪B)⊇s*Cl(A)∪s*Cl(B). (viii) s*Cl(A⋂B)⊆s*Cl(A)⋂s*Cl(B). (ix) Cl(s*Cl(A))=Cl(A). (x) s*Cl(Cl(A))=Cl(A). Proof: (i), (ii), (iii) and (iv) follow from Definition 3.2. (v) follows from Theorem 3.16 and Theorem 3.17. (vi) follows from Theorem 3.18 and Theorem 3.23. (vii) and (viii) follow from (iv) above. From (iii) above, we have A⊆s*Cl(A) and hence Cl(A)⊆Cl(s*Cl(A)). Also from (vi) above, we have Cl(A)⊇s*Cl (A) and hence Cl (A)⊇Cl(s*Cl(A)).Therefore Cl(s*Cl(A))=Cl(A). This proves (ix). Clearly, (x) follows from the fact that Cl (A) is closed and hence semi*-closed and by invoking Theorem 3.17. Remark 3.41: In (vi) of Theorem 3.40, the inclusions may be strict and equality may also hold. This can be seen from the following example. Example 3.42: Let X={a, b, c, d, e, f, g} be endowed with τ={ϕ, {a}, {b}, {a, b}, {a, b, e}, {a, c, d}, {b, f, g}, {a, b, c, d}, {a, b, f, g}, {a, b, c, d, e}, {a, b, e, f, g}, {a, b, c, d, f, g}, X}.Here 𝓕={ϕ, {e}, {c, d}, {f, g}, {c, d, e}, {e, f, g}, {a, c, d, e}, {b, e, f, g}, {c, d, f, g}, {c, d, e, f, g}, {a, c, d, e, f, g}, {b, c, d, e, f, g}, X}; SC(X, τ)={ϕ, {c}, {d}, {e}, {f}, {g},{c, d}, {c, e}, {c, f}, {c, g}, {d, e}, {d, f}, {d, g}, {e, f}, {e, g}, {f, g}, {a, c, d}, {b, f, g}, {c, d, e}, {c, d, f}, {c, d, g}, {c, e, f}, {c, e, g}, {c, f, g}, {d, e, f}, {d, e, g}, {d, f, g}, {e, f, g}, {a, c, d, e}, {a, c, d, f}, {a, c, d, g}, {b, c, f, g}, {b, d, f, g}, {b, e, f, g}, {c, d, e, f}, {c, d, e, g}, {c, d, f, g}, {c, e, f, g}, {d, e, f, g}, {a, c, d, e, f}, {a, c, d, e, g}, {a, c, d, f, g}, {b, c, d, f, g}, {b, c, e, f, g}, {b, d, e, f, g}, {c, d, e, f, g}, {a, c, d, e, f, g}, {b, c, d, e, f, g}, X}; S*C(X, τ)={ϕ, {e}, {c, d}, {f, g}, {a, c, d}, {b, f, g}, {c, d, e}, {e, f, g}, {a, c, d, e}, {b, e, f, g}, {c, d, f, g}, {a, c, d, f, g}, {b, c, d, f, g}, {c, d, e, f, g}, {a, c, d, e, f, g}, {b, c, d, e, f, g}, X}. Let A={e, f, g}. Then Cl(A)=s*Cl(A)=sCl(A)=A. Let B={b, c, d, f, g}. Then Cl(B)={b, c, d, e, f, g}; s*Cl(B)=sCl(B)={b, c, d, f, g}. Here B=sCl(B)=s*Cl(B)⊊Cl(B). Let C={f}. Then sCl(C)={f}; s*Cl(C)=Cl(C)={f, g}. Here C=sCl(C)⊊s*Cl(C)=Cl(C). Let D={a, c, e}. Then sCl(D)=s*Cl(D)=Cl(D)={a, c, d, e}. Here D ⊊sCl(D)=s*Cl(D)=Cl(D) Let E={b, d, f, g}. Then sCl(E)={b, d, f, g}; s*Cl(E)={b, c, d, f, g}; Cl(E)={ b, c, d, e, f, g}. Here E=sCl(E)⊊s*Cl(E)⊊Cl(E). Let F={a, f, g}. Then sCl(F)=s*Cl(F)={a, c, d, f, g}; Cl(F)={ a, c, d, e, f, g}. Here F⊊sCl(F)=s*Cl(F)⊊Cl(F). Let G={b, c, e, g}. Then sCl(G)={b, c, e, f, g}; s*Cl(G)=Cl(G)={b, c, d, e, f, g}. Here G⊊sCl(G)⊊s*Cl(G)=Cl(G). Let H={a, g}. Then sCl(H)={a, c, d, g}; s*Cl(H)={a, c, d, f, g}; Cl(H)={ a, c, d, e, f, g}. Here H⊊sCl(H)⊊s*Cl(H)⊊Cl(H). Remark 3.43: The inclusions in (vii) and (viii) of Theorem 3.40, may be strict and equality may also hold. This can be seen from the following examples. Example 3.44: Consider the space (X, τ) in Example 3.42. Let A={a, d} and B={d, e} then A∪B={a, d, e} and A⋂B={d}. s*Cl(A)={a, c, d}; s*Cl(B)={c, d, e}; s*Cl(A∪B)={a, c, d, e}; s*Cl(A⋂B)={c, d}. Here s*Cl (A∪B)=s*Cl (A)∪s*Cl(B) and s*Cl(A⋂B)=s*Cl (A)⋂s*Cl (B). Let C={b, f } and D={a, c} then C∪D={a, b, c, f}; s*Cl (C)={b, f, g}; s*Cl (D)={a, c, d}; s*Cl(C∪D)=X ; s*Cl(C)∪s*Cl(D)={a, b, c, d, f, g}. Here s*Cl(C)∪s*Cl(D)⊊s*Cl(C∪D). Let E={a, e} and F={c, e, f} then E⋂F={e}; s*Cl(E)={a, c, d, e}; s*Cl(F)={c, d, e, f, g}; s*Cl(E⋂F)={e}; s*Cl(E)⋂s*Cl(F)={c, d, e}.Here s*Cl(E⋂F)⊊ s*Cl(E)⋂s*Cl(F). Theorem 3.45: Let ß be a collection of subsets in (X, τ) satisfying (i) 𝓕⊆ß where 𝓕 is the collection of closed sets in (X, τ) (ii) If B∈ß and D⊆X such that Int*(B)⊆D⊆B implies D∈ß. Then S*C(X, τ)⊆ß. Thus S*C(X, τ) is the smallest collection satisfying the (i) and (ii). Proof: Let A∈S*C(X, τ).Then by Theorem 3.7, there is a closed set F in (X, τ) such that Int*(F)⊆A⊆F. By (i), F∈ß. By (ii), A∈ß. Hence S*C(X, τ)⊆ß. By using Theorem 3.18 and Theorem 3.10, S*C(X, τ) satisfies (i) and (ii).Thus S*C(X, τ) is the smallest collection satisfying (i) and (ii). Theorem 3.46: Let A⊆X and let x∈X. Then x∈s*Cl(A) if and only if every semi*-open set in X containing x intersects A. Proof: Sufficiency. Suppose x∉s*Cl(A). Then X\s*Cl(A) is a semi*-open set containing x that does not intersect A. Necessity. Suppose U is a semi*-open set containing x that does not intersect A. Then X\U is a semi*-closed set containing A. Therefore s*Cl(A)⊆X\U. Hence x∉s*Cl(A). Thus x∉s*Cl(A) if and only if there is a semi*open set containing x that does not intersect A. This proves the theorem. 175 Robert et.al/On Semi*- Closed Sets Corollary 3.47: Let A be a subset of X. s*Cl(A) is the set of all semi*-adherent points of A. Proof: Follows from Definition 3.3. Theorem 3.48: If A is a subset of X, then (i) s*Cl(X\A)=X\s*Int(A). (ii) s*Int(X\A)=X\s*Cl(A). Proof: (i) Let x∈X\s*Int(A).Then x∉s*Int(A).This implies that x does not belong to any semi*-open subset of A. Let F be a semi*-closed set containing X\A. Then X\F is a semi*open set contained in A. Therefore x∉X\F and so x∈F. Hence x∈s*Cl(X\A). Therefore X\s*Int(A)⊆ s*Cl(X\A). On the other hand, let x∈s*Cl(X\A). Then x belongs to every semi*-closed set containing X\A. Hence x does not belong to any semi*-open subset of A. That is x∉s*Int(A). Then x∈X\s*Int(A).This proves (i). (ii) can be proved by replacing A by X\A in (i) and using set theoretic properties. Theorem 3.49: If A⊆X, then s*Cl(A)=A∪Ds*[A]. Proof: By definition, A⊆s*Cl(A). If x∈Ds*[A], then by Theorem 3.47 and Definition 3.5, x∈s*Cl(A). Thus A∪ Ds*[A]⊆s*Cl(A). On the other hand, let x∈s*Cl(A). If x∈A, then x∈A∪Ds*[A]. Suppose x∉A. We claim that x is a semi*limit point of A. Let U be a semi*-open set containing x. Then U intersects A. Otherwise X\U is a semi*-closed set containing A and since x∈s*Cl(A), we have x∈X\U. This is a contradiction to x∈U. Hence x∈Ds*[A]. Therefore s*Cl(A)⊆A∪Ds*[A]. Hence the theorem. Corollary 3.50: A subset A of X is semi*-closed if and only if A contains all its semi*-limit points. That is, A is semi*closed if and only if Ds*[A]⊆A. Proof: Follows from Theorem 3.17 and Theorem 3.49. Theorem 3.51: If A is a subset of a topological space X, (i) s*Cl(A)=A∪Int*(Cl(A)). (ii) s*Int(A)=A⋂Cl*(Int(A)). Proof: Int*(Cl(A∪Int*(Cl(A))))=Int*(Cl(A)∪Cl(Int*(Cl(A)))) =Int*(Cl(A))⊆A∪Int*(Cl(A)). Then by Theorem 3.9, A∪Int*(Cl(A)) is a semi*-closed set containing A. Hence s*Cl(A) ⊆A∪Int*(Cl(A)). Also since s*Cl(A) is semi*-closed , by invoking Theorem 3.9 again, we get Int*(Cl(A))⊆Int*(Cl(s*Cl(A)))⊆s*Cl(A). Hence A∪Int*(Cl(A))⊆s* Cl(A) which proves (i). (ii) follows from (i) above by replacing A by X\A and applying Theorem 3.48. Theorem 3.52: If a subset A of a space X is nowhere dense, then Int(s*Cl(A))=. Proof: Follows from Int(Cl(A))= and the fact Int(s*Cl(A))⊆Int(Cl(A)). REFERENCES [1] Biswas,N., On Characterization of Semi-Continuous Functions, Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 48(8)1970, 399-402. [2] Crossley, S.G and Hildebrand, S.K., Semi-closure, Texas Journal of Sc. 22 (1971), 99-112. [3] Das,P., Note on Some Applications of Semi-Open Sets, Prog. Math. 7 (1973), 33-44. [4] Di Maio,G. and Noiri,T., On s-closed spaces, Indian Journal of Pure and Applied Mathematics. 18(3)(1987 ), 226-233. [5] Dunham,W., A New Closure Operator for Non-T1 Topologies, Kyungpook Math. J. 22 (1982), 55-60. [6] Khalimsky,E.D, Applications of Connected Ordered Topological spaces in Topology, Conference of Math. Departments of Povolsia, 1970. [7] Levine,N., Semi-Open Sets and Semi-Continuity in Topological Spaces, Amer. Math. Monthly. 70 (1963), 36-41 [8] Levine,N., Generalized Closed Sets in Topology, Rend. Circ. Mat. Palermo. 19(2) (1970), 89-96. [9] Mashhour, A.S., Abd El-Monsef, M.E. and El-Deeb, S.N., On Precontinuous and weak precontinuous mappings. Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53. [10] Njastad, O., On Some Classes of Nearly Open Sets, Pacific J. Math. 15(1965) No. (3), 961-970. [11]Robert, A., and Pious Missier, S. A New Class of Nearly Open Sets, International Journal of mathametical archive. [12]Willard, S., General Topology, Addition Wesley (1970) 176