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Indian Journal of Mathematics and Mathematical Sciences Vol. 7, No. 2, (December 2011) : 161-167 ON sp-gpr-COMPACT AND sp-gpr-CONNECTED IN TOPOLOGICAL SPACES D. Christia Jebakumari & M. Mariasingam Abstract Compact spaces and connected spaces constitute the most important classes of topological spaces. They find very active role in abstract analysis. In this paper, by using sp-gpr-closed and sp-gpr-open sets, we introduce two concepts “sp-gpr-compact” and “sp-gpr-connected” in topological spaces. We study their relations with other compact and connected spaces. Keywords: sp-gpr-open, sp-gpr-compact, sp-gpr-connected etc. 1. INTRODUCTION AND PRELIMINARIES Levine [7] introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. Using pgpr-closed sets and pgpr-open sets, Anitha and Thangavelu [3] defined and studied pgpr-compact and pgpr-connected spaces. In this paper we have introduced sp-gpr-compact and sp-gpr-connected spaces. Using these new types of spaces, several characterizations and its properties have been obtained.Throughout this paper (X, τ), (Y, σ) denote the topological spaces. For any subset A of (X, σ) cl A, int A denote the closure of A and interior of A respectively. We recall some definitions and results that are useful in the sequel. Definition 1.1: A subset A of a space (X, τ) is regular open [10] if A = int cl A and regular closed if A = cl int A; pre-open [8] if A ⊆ int cl A and pre-closed if cl int A ⊆ A; semi-pre-open [1] if A ⊆ cl int cl A and semi-pre-closed if int cl int A ⊆ A. The pre-closure (resp. semi-pre-closure) of a subset A of X is the intersection of all pre-closed (resp. semi-pre-closed) sets containing A and is denoted by pcl A (resp. spcl A). The semi-pre-interior of a subset A of X is the union of all semi-pre-open sets contained in A and is denoted by sp int A. Definition 1.2: A subset A of (X, τ) is (i) g-closed [7] if cl A ⊆ U, whenever U is an open set containing A. (ii) rg-closed [9] if cl A ⊆ U whenever U is a regular open set containing A. (iii) pre-generalized pre-regular closed (briefly pgpr-closed) [2] if pcl A ⊆ U whenever A ⊆ U and U is rg-open. 162 D. Christia Jebakumari & M. Mariasingam (iv) semi-pre-generalized pre- regular closed (briefly sp-gpr-closed) [5] if spcl A ⊆ U whenever A ⊆ U and U is rg-open. A subset B of a space X is g-open if X \B is g-closed. The concepts of rg-open, pgpr-open, sp-gpr-open are analogously defined. Definition 1.3 [5]: A function f : (X, τ) → (Y, σ) is said to be (i) sp-gpr-continuous if f – 1(V) is sp-gpr-closed subset of X for every closed set V in Y. (ii) sp-gpr-irresolute if f – 1(V ) is sp-gpr-closed subset of X for every sp-gpr-closed subset V in Y. Definition 1.4 [3]: A topological space X is pgpr-compact if every cover of X by pgpr-open subsets of X has a finite sub cover. Definition 1.5 [3]: A topological space (X, τ) is said to be pgpr-connected if X cannot be written as the disjoint union of two non-empty pgpr-open sets in X. Lemma 1.6 [5]: (i) Every closed set is sp-gpr-closed. (ii) Every open set is sp-gpr-open. (iii) Every pgpr-closed set is sp-gpr-closed. (iv) Every pgpr-open set is sp-gpr-open. Lemma 1.7 [5]: A set A ⊆ X is sp-gpr-open if and only if F ⊆ sp int A whenever F ⊆ A, F is rg-closed. Lemma 1.8 [3]: For a topological space X, the following are equivalent: (i) X is pgpr-connected. (ii) The only subsets of X which are both pgpr-open and pgpr-closed are the empty set φ and X. (iii) Each pgpr-continuous function of X into a discrete space Y with at least two points is a constant map. Lemma 1.9 [9]: Suppose that B ⊆ A ⊆ X, B is rg-closed relative to A and that A is a g-closed open subset of X. Then B is rg-closed relative to X. 2. sp-gpr-COMPACT SPACE Definition 2.1: A topological space X is sp-gpr-compact if every cover of X by sp-gpr-open subsets of X has a finite sub cover. On sp-gpr-Compact and sp-gpr-Connected in Topological Spaces 163 Definition 2.2: A subset S of a topological space X is sp-gpr-compact relative to X ifΑfor every collection {Aα : α ∈ Ω} of sp-gpr-open subsetsΑof X such that S ⊆ α there exists a finite subset ∆ of Ω such that S ⊆ α . α∈Ω α∈∆ Theorem 2.3: A sp-gpr-closed subset of a sp-gpr-compact space X is sp-gpr-compact relative to X. Proof: Let A be sp-gpr-closed subset of a sp-gpr-compact space X. Then X \ A is sp-gpr-open. Let ξ be a cover for A by sp-gpr-open subsets of X. Then ξ ∪ {X \A} is a cover for X by sp-gpr-open subsets of X. Since X is sp-gpr-compact, by Definition 2.1 it has a finite sub cover, say {P1, P2, …, Pn} = ℘. If X \ A ∈ ℘, then ℘ is a finite sub cover of ξ for A. If X \ A ∈ ℘, then ℘ \ {X \ A} is a finite sub cover of ξ for A. Thus A is sp-gpr-compact relative to X. Theorem 2.4: Let f : (X, τ) → (Y, σ) be a surjective, sp-gpr-continuous function. If X is sp-gpr-compact, then Y is compact. Proof: Suppose X is sp-gpr-compact. Let {Aα : α ∈ Ω} be an open cover of Y. Since f is sp-gpr-continuous by Definition 1.3(i) {f –1(Aα) : α ∈ Ω} is a cover for X by sp-gpr-open sub sets of X. Since X is sp-gpr-compact by Definition 2.1 it has a finite sub cover say {f –1(A1), f –1(A2), …, f –1(An)}. Since f is a surjection, {A1, A2, …, An} is an open cover of Y and hence Y is compact. This proves the theorem. Theorem 2.5: Suppose f : (X, τ) → (Y, σ) is sp-gpr-irresolute. Let S ∈ X be sp-gpr-compact relative to X. Then the image f (S) is sp-gpr-compact relative to Y. Proof: Let {Aα : α ∈ Ω} be a collection of sp-gpr-open sets in Y such that Α Α –1 f ( S ) ⊆ α . Then S ⊆ f ( α) . Since f is sp-gpr-irresolute, by α∈Ω α∈Ω Definition 1.3(ii), f –1(Aα) is sp-gpr-open in X for each α. Since S is sp-gprcomptact relative to X, by Definition 2.2 there exists a finite sub collection Α Α {f –1(A1), f –1(A2), …, f –1(An)} such that S ⊆ n α =1 f −1 ( α ) . That is, f ( S ) ⊆ n α =1 α . Hence by Definition 2.2 f (S) is sp-gpr-compact relative to Y. Theorem 2.6: Every sp-gpr-compact space is pgpr-compact. Proof: Let {Aα : α ∈ Ω} be a collection of pgpr-open sets which covers X. By Lemma 1.6(iii) each Aα is sp-gpr-open. Since X is sp-gpr-compact by Definition 2.1 there exists a finite sub-cover X. By Definition 1.4 X is pgpr-compact. Theorem 2.7: Let (X, τ) be a topological space. X is sp-gpr-compact if and only if any family of sp-gpr-closed sets with finite intersection property has non-empty intersection. 164 D. Christia Jebakumari & M. Mariasingam Proof: Suppose X is sp-gpr-compact. Let {Aα} be a family of sp-gpr-closed subsets of X with finite intersection property. We claim that Αα ≠ φ . Suppose Αα = φ . Α X α α Then X \ α = . Therefore, ( X \ Aα ) = X . Also, since each Aα is sp-gpr-closed, ( α ) α X \ Aα is sp-gpr-open. Therefore, {X \ Aα} is a cover for X by sp-gpr-open subsets of X. Since X is sp-gpr-compact, by Definition 2.1 this cover has a finite n subcover Αsay {XX\ A1, X \ A2, …, X \ An}. Therefore, ( Χ \ Αα ) = X . Therefore, n α =1 n X \ α = which implies Αα = φ . Which is a contradiction to the finite α =1 α =1 intersection property. Therefore, Αα ≠ φ . Conversely suppose that each family α of sp-gpr-closed sets in X with finite intersection property has non-empty intersection. We wish to prove that X is sp-gpr-compact. Let {Aα : α ∈ I} be a cover of X by sp-gpr-open subsets of X. Then, Aα = X , that implies X \ Aα is α∈I α∈I Α an empty set. Therefore, ( X \ α ) = φ . Since Aα is sp-gpr-open, X\ Aα is sp-gprα∈I closed for each a Therefore, {X \ Aα : α ∈ I} is a family of sp-gpr-closed sets whose Α intersection is empty. Hence by hypothesis there exists a finite sub collection of n sp-gpr-closed subsets of X say {X \ A1, X \ A2, …, X \ An} such that ( X \ α ) = φ , α =1 n n that implies X \ Αα = φ , which implies Αα = X . This proves that, X is α =1 α =1 sp-gpr-compact. 3. sp-gpr-CONNECTEDNESS Definition 3.1: A topological space (X, τ) is said to be sp-gpr-connected if X cannot be written as the disjoint union of two non-empty sp-gpr-open sets in X. Definition 3.2: A subset S of X is sp-gpr-connected if it is sp-gpr-connected as a subspace that is if S cannot be written as the disjoint union of two non-empty sp-gpr-open sets in S. Definition 3.3: A subset S of a topological space (X, τ) is said to be sp-gprconnected relative to X if S cannot be written as the disjoint union of two non-empty sp-gpr-open sets in X. Theorem 3.4: For a topological space X, the following are equivalent: (i) X is sp-gpr-connected. (ii) The only subsets of X which are both sp-gpr-open and sp-gpr-closed are the empty set φ and X. On sp-gpr-Compact and sp-gpr-Connected in Topological Spaces 165 (iii) Each sp-gpr-continuous function of X into a discrete space Y with atleast two points is a constant map. Proof: (i) (ii): Suppose S ⊆ X is a pre-open subset which is both sp-gpropen and sp-gpr-closed. Then by Definition 1.2 its complement X \ S is also sp-gpr-open and sp-gpr-closed. Then X = S ∪ (X \ S), a \ disjoint union of two non-empty sp-gpr-open sets which contradicts (i). Hence S = φ or X. (ii) (i): Suppose X is not sp-gpr-connected. Then X = A ∪ B where A ∩ B = φ, A ≠ φ, B ≠ φ and A and B are sp-gpr-open. Since A = X \ B, A is sp-gpr-closed. If (ii) holds A = φ or X. A = φ is not possible by our choice. If A = X then B = φ that is also not possible by our choice. This proves that X is sp-gpr-connected. (ii) (iii): Let f : X → Y be a sp-gpr-continuous function where Y is a discrete space with atleast two points. Then f –1({y}) is sp-gpr-closed and sp-gpr-open for each y ∈ Y. By (ii) f –1({y}) = φ or X. If f –1({y}) = φ for all y ∈ Y, f will not be a function. So f –1({y}) = X. This proves that f is constant. (iii) (ii): Let S be both sp-gpr-open and sp-gpr-closed in X. Suppose S ≠ φ. Let a, b in Y and a ≠ b. Let Y be a discrete space. Fix y0 and y1 in Y and y0 ≠ y1. Define f : X → Y by f (x) = y0 for x ∈ S and f (x) = y1 for x ∈ X \ S. Then f is sp-gprcontinuous function. By (iii), f is constant. Therefore f = y0 or f = y1 . If f = y0 then S = X and if f = y1 then S = φ. This proves the theorem. Theorem 3.5: (i) Let f : X → Y be sp-gpr-continuous and onto and X be sp-gpr-connected. Then Y is connected. (ii) Let f : X → Y be sp-gpr-irresolute and onto and X be sp-gpr-connected. Then Y is sp-gpr-connected. Proof: Suppose Y is not connected. Then Y = A ∪ B where A ∩ B = φ, A ≠ φ, B ≠ φ and A and B are open in Y. If f is sp-gpr-continuous and onto, X = f –1(A) ∪ f –1(B) where f –1(A) and f –1(B) are disjoint non-empty sp-gpr-open subsets of X. This contradicts the fact that X is sp-gpr-connected. Thus (i) is proved. Suppose Y is not sp-gpr-connected. Then Y = A ∪ B where A ∩ B = φ, A ≠ f, B ≠ f and A and B are sp-gpr-open in Y. Since f is sp-gpr-irresolute and onto, X = f –1(A) ∩ f –1(B) where f –1(A) and f –1(B) are disjoint non-empty sp-gpr-open subsets of X. This contradicts the fact that X is sp-gpr-connected. Thus (ii) is proved. Theorem 3.6: Every sp-gpr-connected space is connected. Proof: Let X be a sp-gpr-connected space. Then by Theorem 3.4 the only subsets of X which are both sp-gpr-open and sp-gpr-closed are the empty set φ and X. Suppose X is not connected. Then there exist a proper non-empty subset B of X which is both open and closed in X. Since by Lemma 1.6(i)every closed set is 166 D. Christia Jebakumari & M. Mariasingam sp-gpr-closed, B is a proper non-empty subset of X which is both sp-gpr-open and sp-gpr-closed in X. Then by Theorem 3.4, X is not sp-gpr-connected. This proves the theorem. The following example shows that the converse is not true. Example 3.7: Let X = {a, b, c}, τ = {φ, {a}, {b}, {a, b}, X }. The topological space (X, τ) is connected. However, since {a, c} and {b} are both sp-gpr-open but X is not sp-gpr-connected. Theorem 3.8: Every sp-gpr-connected space is pgpr-connected. Proof: Let X be a sp-gpr-connected space. By Theorem 3.4 the only subsets of X which are both sp-gpr-open and sp-gpr-closed are the empty set φ and X. Suppose X is not pgpr-connected. Then by Lemma 1.8 there exist a proper non-empty subset B of X which is both pgpr-open and pgpr-closed in X. By Lemma 1.6(iii) B is both sp-gpr-open and sp-gpr-closed in X. Then by Theorem 3.4 X is not sp-gpr-connected. This proves the Theorem. The following example shows that the converse is not true. Example 3.9: Let X = {a, b, c}, τ = {φ, {a}, {b}, {a, b}, X}. The topological space (X, τ) is pgpr-connected. However, since {a, c} and {b} are both sp-gpropen but X is not sp-gpr-connected. Lemma 3.10: Suppose Y ⊆ X, Y is clopen. If C is sp-gpr-open in X, then C ∩ Y is sp-gpr-open in Y. Proof: Let F ⊆ C ∩ Y and F be rg-closed in Y. Since Y is g-closed and open relative to X, by Lemma 1.9 F is rg-closed in X. Since F ⊆ C, by Lemma 1.7 F ⊆ sp int (C) that implies F ⊆ Y ∩ sp int (C) = sp intY (C ∩ Y ). Again by Lemma 1.7, C ∩ Y is sp-gpr-open in Y. Theorem 3.11: Let Y ⊆ X be clopen in X. Suppose X = C ∪ D where C and D are two disjoint non-empty sp-gpr-open subsets of X. If Y is a sp-gpr-connected subspace of X, then Y lies entirely within either C or D. Proof: Since C and D are both sp-gpr-open in X, by Lemma 3.10 the sets C ∩ Y and D ∩ Y are sp-gpr-open in Y. These two sets are disjoint and their union is Y. If they were both non-empty, then Y is not sp-gpr-connected. Therefore, one of them is empty. Hence Y must lie entirely in C or in D. Theorem 3.12: Let {Aα : α ∈ Ω} be a locally finite family of clopen sets in X such that they have a common point. If each Aα is a sp-gpr-connected subspace of X then their union is a sp-gpr-connected subspace of X. Proof: Let p be a point of ∩Aα. Let Y = ∪Aα. Then Y is clopen. Suppose that Y = C ∪ D, where C and D are two disjoint non-empty sp-gpr-open subsets of Y. On sp-gpr-Compact and sp-gpr-Connected in Topological Spaces 167 The point p is in one of the sets C or D. If p ∈ C then Aα ⊆ C for every a, so that ∪ Aα ⊆ C. This shows that D = φ. Therefore Y = C is sp-gpr-connected. Theorem 3.13: Let (X, τ) and (Y, σ) be any topological spaces. Let f : X → Y be a sp-gpr-irresolute function and X be sp-gpr-connected subset of Y. Then f (X) is sp-gpr-connected subset of Y. Proof: Let f (X) = S so that f is a function from X onto S. Suppose S is not sp-gpr-connected. Then there is a proper non-empty subset B of which is both sp-gpr-open and sp-gpr-closed in S. Since f is sp-gpr-irresolute, by Definition1.3(ii) f –1(B) is proper non-empty subset of X which is both sp-gpr-open and sp-gpr-closed in X. Then Theorem3.4, X is not sp-gpr-connected. This shows that S is sp-gpr-connected. REFERENCES [1] D. Andrijevic, (1986), Semi-Preopen Sets, Mat. Vesnik, 38, 24-32. [2] M. Anitha, and P. Thangavelu, (2005), On pre-Generalized pre-Regular Closed Sets, Acta Ciencia Indica, 31M(4), 1035-1040. [3] M. Anitha, and P. Thangavelu, (2008), On pgpr-Compact and pgpr-Connected Spaces Dissertation, 45-55. [4] P. Bhattacharya, and B. K. Lahiri, (1987), Semi-Generalized Closed Sets in Topology, Indian J. Math., 29(3), 375-385. [5] D. Christia Jebakumari, M. Mariasingam, and P. Thangavelu, (2011), Presented a Paper On sp-gpr-Closed Sets in International Seminar on New Trends in Applications of Mathematics (ISNTAM), Bharata Mata College, Thrikkakara, Kochi, India, January 31, February 01-02 (2011). [6] Y. Gnanambal, (1997), On Generalized pre-Regular Closed Sets in Topological Spaces, Indian J. Pure Appl. Math., 28(3), 351-360. [7] N. Levine, (1970), Generalized Closed Sets in Topology, Rend. Circ, Mat. Palermo, 19(2), 89- 96. [8] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb, (1982), On Precontinuous and Weak Precontinuous Functions, Proc.Math.Phys.Soc.Egypt, 53, 47-53. [9] N. Palaniappan, and K. C. Rao, (1993), Regular Generalized Closed Sets, Kyungpook Math. J., 33, 211-219. [10] M. H. Stone, (1937), Applications of the Theory of Boolean Rings to the General Topology, Trans. A.M.S., 41, 375-481. D. Christia Jebakumari Department of Mathematics, Sarah Tucker College, Tirunelveli-627007, India. E-mail: [email protected] M. Mariasingam Department of Mathematics, V.O. Chidambaram College, Thoothukudi-628008, India.