K. Chandrasekhara Rao and D. Narasimhan. Department of Mathematics, Srinivasa Ramanujan Centre. SASTRA University, Kumbakonam - 612 001, India.
International Mathematical Forum, 4, 2009, no. 21, 1033 - 1037
Characterizations of TS - Spaces K. Chandrasekhara Rao and D. Narasimhan Department of Mathematics, Srinivasa Ramanujan Centre SASTRA University, Kumbakonam - 612 001, India k.chandrasekhara@rediffmail.com dnsastra@rediffmail.com Abstract The aim of this paper is to study the characterizations of TS - spaces in general topology.
Mathematics Subject Classification: 54A05 Keywords: TS - space, space,g Tδg - space
1
α Tc
- space,
α Tb
∗ - space, Tαgs - space, T1/2 -
Introduction
Levine15,16 introduced semi open sets and g - closed sets. Njastad17 introduced α - open sets. Abd El-Monsef et.al1 invented β - open sets. Andrijevic3 called β - open sets as semi pre open sets. Mashhour et.al 2 formulated pre open sets. Maki et.al8,9 introduced αg - closed sets and gα - closed sets. Bhattacharya and Lahiri5 , Arya and Nour4 , Duntchev13 , Gnanambal14 and Chandrasekhara Rao and Joseph6 investigated sg - closed sets, gs - closed sets, gsp closed sets, gpr - closed sets and s∗ g - closed sets respectively. Dunham12 , Bhattacharya and Lahiri5 , Duntchev13 , Gnanambal14 were familiar with T1/2 , semi - T1/2 , semi pre - T1/2 and pre regular T1/2 - spaces respectively. Devi et.al10,11 introduced Tb , Td and α Tb , α Td - spaces respectively. Veera Kumar18−20 in∗ , ∗ T1/2 , Tp∗ , ∗ Tp - spaces. troduced T1/2 Chandrasekhara Rao and Narasimhan7 introduced TS - spaces. In this paper we study the characterizations of TS - spaces.
2
Preliminaries
Let (X, τ ) or simply X denote a topological space. For any subset A ⊆ X , the closure [ resp. δ - closure, α - closure ] of a subset A of a space (X, τ ) is
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the intersection of all closed [ resp. δ - closed, α - closed ] sets that contain A and is denoted by cl(A) [resp. clδ (A), αcl(A)]. We shall require the following known definitions. Definition 2.1 A set A of a topological space (X, τ ) is called (a) semi open if there exists an open set U such that U ⊆ A ⊆ cl(U), (b) semi closed if X − A is semi open. equivalently, a set A of a topological space (X, τ ) is called semi closed if there exists a closed set F such that int(F ) ⊆ A ⊆ F , (c) generalized closed (g - closed ) if cl (A) ⊆ U whenever A ⊆ U and U is open in X, (d) generalized open (g - open ) if X − A is g - closed, (e) generalized semi open (gs - open) if F ⊆ sint (A) whenever F ⊆ A and F is closed in X, (f) generalized semi closed (gs - closed) if X − A is gs - open, (g) semi star generalized closed (s∗ g - closed ) if cl ( A) ⊆ U whenever A ⊆ U and U is semi open in X, (h) semi star generalized open (s∗ g - open ) if X − A is s∗ g - closed in X, (i) α−open if A ⊆ int { cl [ int(A)]}, (j) α−closed if cl { int [ cl(A)] }⊆ A, (k) αg- closed if αcl (A) ⊆ U whenever A ⊆ U and U is open in X, (l) αgs- closed if αcl (A) ⊆ U whenever A ⊆ U and U is semi open in X, (m) g ∗ - closed if cl (A) ⊆ U whenever A ⊆ U and U is g - open in X, (n) δ − g closed if clδ (A) ⊆ U whenever A ⊆ U and U is open in X, (o) δ − g closed if clδ (A) ⊆ U whenever A ⊆ U and U is semi open in X. Definition 2.2 A topological space (X, τ ) is called (a) T1/2 - space if every g - closed set is closed, ∗ (b) T1/2 - space if every g ∗ - closed set is closed,
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(c) Tc - space if every gs - closed set is g ∗ - closed, (d) α Tb - space if every αg - closed set is closed, (e) g Tδ g - space if every g - closed set is δ − g - closed, (f) α Tc - space if every αg - closed set is g ∗ - closed, (g) complemented space if every open set is closed, (h) Tαgs - space if every αgs - closed set is closed.
3
- Characterizations of TS - Spaces
Definition 3.1 A topological space (X, τ ) is called TS - space if every s∗ g closed set is closed in X. Example 3.2 Let X = {a, b, c}, τ = {φ, X, {a}}. Then s∗ g - closed sets in X are φ, X, {b, c}, which are closed in X. Hence (X, τ ) is TS - space. Theorem 3.3 A topological space (X, τ ) is TS - space if and only if the singleton x is either open or semi closed. Proof. Let X be a TS - space and suppose that {x} is not semi closed. Then X - {x} is not semi open. Consequently X is the only semi open set containing the set X - {x}. Therefore, X − {x} is s∗ g - closed in X. Since X is a TS space, we have X − {x} is closed in X. Consequently, {x} is open in X. Conversely, suppose that {x} is either open or semi closed. Let A be a s∗ g closed in X. Obviously A ⊆ cl (A). Let x ∈ cl (A) Case i: Suppose that {x} is open. Since x ∈ cl (A), we have x ∈ A.Thus, cl (A) ⊆ A. Case ii: Suppose that {x} is semi closed and x does not belongs to A. Then cl (A) - A contains the semi closed set {x}. This is a contradiction to the fact that A is s∗ g - closed in X. Hence, x ∈ A, implies that cl (A) ⊆ A. Therefore cl (A) = A. Hence X is a TS - space. Theorem 3.4 If X is complemented TS - space, then X is a T1/2 - space Proof. Let A be a g - closed set in X. Let A ⊆ U and U is semi open in X. Since X is a complemented space, we have U is open in X. Since A is g - closed, we have cl(A) ⊆ U. Hence A is s∗ g - closed in X.Since X is a TS space, we have A is closed in X. Hence X is a T1/2 - space. ∗ - space then Theorem 3.5 If a topological space (X, τ ) is TC - space and T1/2 X is a TS - space.
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Proof. Let A be a s∗ g - closed set in X. Then A is a gs - closed set in X. ∗ Since X is a TC - space, we have A is g ∗ - closed in X. Since X is a T1/2 space, we have A is closed in X. Hence X is a TS - space. Theorem 3.6 In any topological space (X, τ ) , every δ − g closed set is s∗ g closed. Proof. Let A be a δ − g closed set in X. Let A ⊆ U and U is semi open in X.Since A is δ − g closed in X, we have clδ (A) ⊆ A. Since cl(A) ⊆ clδ (A), we have cl(A) ⊆ U. Hence A is s∗ g- closed in X. g Remark 3.7 Since every s∗ g - closed set is closed in a TS - space, every δ − closed set is closed in a TS - space. Theorem 3.8 If a topological space (X, τ ) is TS - space and g Tδ g - space then X is a T1/2 - space. Proof. Let A be a g - closed set in X. Since X is a g Tδ g - space, we have A is δ − g closed in X. Consequently A is s∗ g - closed in X. Since X is a TS space, we have A is closed in X. Hence X is a T1/2 - space. Theorem 3.9 If X is a complemented space, then every δ − g closed set is s∗ g - closed in X. ∗ - space. Theorem 3.10 Every α Tc - space X is a TS - space if X is T1/2
Proof. Let X be a α Tc - space. Then every αg - closed set is g ∗ - closed in X. ∗ - space, we have every g ∗ - closed set is closed in X. Hence Since X is a T1/2 every αg - closed set is closed in X. Consequently, X is a α Tb - space. Since every α Tb - space is a TS - space, we have X is a TS - space. Theorem 3.11 Every Tαgs - space X is a TS - space.
References [1] Abd El - Monsef, M.E., El - Deep S.N. & Mahmond, R.A. (1983) Bull. Fac. Sec. Assiut univ., 12 : 77 - 90. [2] Abd. El - Monsef, M.E. & El - Deep,S.N., Mashhour,A.S.,(1982)Proc. Math. and Phys. Soc. Egypt ., 53 : 47 - 53. [3] Antrijevic, D. (1986) Mat. Vesnik, 38(1) : 24 - 32.
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[4] Arya, S.P. & Nour,T. (1990) Indian J. pure.Appl. Math., 21(8) : 717 719. [5] Bhattacharya, P. & Lahari, B.K. (1987) Indian J. Math., 29(3) :375 - 382. [6] Chandrasekhara Rao K. & Joseph, K.(2000) Bulletin of Pure and Applied Sciences, 19 E ( 2 ) : 281 - 290. [7] Chandrasekhara Rao, K & Narasimhan, D, (2007) Proc. Nat. Acad. Sci. India, 77(A), IV:363 - 366. [8] Devi,R., Balachandran K.& Maki, H.(1993) Fukuoka Univ. Ed . Part II, 42:13 21. [9] Devi,R.,Balachandran K. & Maki, H.(1993) Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 14 : 41 - 54. [10] Devi,R., Balachandran K. & Maki, H.(1994) Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 15 : 51 - 63. [11] Devi,R., Balachandran K. & Maki, H. (1998) Indian J. pure. Appl. Math., 29(1) : 37 - 49. [12] Dunham, W.(1997) Kyungpook Math. J., 17 : 161 - 169. [13] Duntchev, J. (1995) Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 16: 35 48. [14] Gnanambal,Y. (1997) Indian J. pure. Appl. Math., 28(3) : 351 - 360. [15] Levine,N. (1963) Amer. Math. Monthly, 70 : 36 - 41. [16] Levine, N. (1970) Rend. Circ. Math. Palermo, 19 ( 2 ) : 89 - 96. [17] Njastad, O. ( 1965 ) Pacific J. Math., 15 : 961 - 970. [18] Veera Kumar, M.K.R.S. (2000) Mem. Fac.Sci. Kochi Univ.Ser. A. Math.,21:1-19. [19] Veera Kumar ,M.K.R.S. (2002) Acta Ciencia Indica (Mathematics) Meerut , XX VIII , M ( 1 ) : 51 - 60. [20] Veera Kumar , M.K.R.S. (2004) Antartica J. Math., 1(1) : 9 - 16. Received: July, 2008
