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126 Md. Sanam Suraj et al17. Boutis T. & Papadakis K.E. Stability ofvertical motion in the N-body circular Sitnikovproblem. Celestial Mechanics and DynamicalAstronomy, 104 (1-2): 205-225 (2009).18. T. Kovacs, Gy. Bene & T. Tél, 2011.Relativistic effects in the chaotic Sitnikovproblem. Monthly Notices of RoyalAstronomical Society. 414 (3): 2275-2281(2011).19. McCuskey. C.W. Introduction to CelestialMechanic., Addison-Wesley publicationCompany, Reading, Massachusetts; Palo Alto,London (1962).
Proceedings of the Pakistan Academy of Sciences 48 (2): 127–133, 2011Copyright © Pakistan Academy of SciencesISSN: 0377 - 2969-CLOSED SETS IN TOPOLOGYPakistan Academy of SciencesOriginal ArticleO. Ravi 1* , J. Antony Rex Rodrigo 2 , S. Ganesan 3 and A. Kumaradhas 41 Department of Mathematics, P.M. Thevar College, Usilampatti,Madurai District, Tamil Nadu, India2 Department of Mathematics, V.O. Chidambaram College,Thoothukudi, Tamil Nadu, India3 Department of Mathematics, N.M.S.S.V.N. College,Nagamalai, Madurai, Tamil Nadu, India4 Department of Mathematics, Vivekananda College,Agasteeswaram, Kanyakumari, Tamil Nadu, IndiaAbstract: In this paper, we introduce a new class of sets called-closed sets in topological spaces.We prove that this class lies between α -closed sets and gα -closed sets. We discuss some basicproperties of -closed sets.Keywords: Topological space, sg-closed set, -closed set, -closed set, gp-closed set, gsp-closed set2000 Mathematics Subject Classification: 54C10, 54C08, 54C051. INTRODUCTIONThe concept of generalized closed sets play asignificant role in topology. There are manyresearch papers which deals with different typesof generalized closed sets. Bhattacharya andLahiri [3] introduced sg-closed set in topologicalspaces. Arya and Nour [2] introduced gs-closedsets in topological spaces. Sheik John [16]introduced ω -closed sets in topological spaces.Rajamani and Viswanathan [14] introducedαgs-closed sets in topological spaces. QuiteRecently, Ravi and Ganesan [15] introduced -closed sets and proved that they forms atopology. In this paper we introduce a new classof sets, namely -closed sets, for topologicalspaces and study their basic properties.2. PRELIMINARIESThroughout this paper (X, τ) and (Y, σ) (or Xand Y) represent topological spaces on which noseparation axioms are assumed unless otherwisementioned. For a subset A of a space (X, τ),cl(A), int(A) and A c or X − A denote the closureof A, the interior of A and the complement of Arespectively.−−−−−−−−−−−−−−−−−−−−−−−Received December 2010, Accepted June 2011*Corresponding author: O. Ravi; Email: siingam@yahoo.comWe recall the following definitions which areuseful in the sequel.Definition 2.1A subset A of a space (X, τ) is called:(i) semi-open set [8] if A ⊆ cl(int(A));(ii) preopen set [11] if A ⊆ int(cl(A));(iii) α -open set [12] if A ⊆ int(cl(int(A)));(iv) semi-preopen [1] if A ⊆ cl(int(cl(A))).The complements of the above mentioned opensets are called their respective closed sets.The preclosure [13] (resp. semi-closure [5],α -closure [12], semi-pre-closure [1]) of asubset A of X, denoted by pcl(A) (resp. scl(A),α cl(A), spcl(A)) is defined to be theintersection of all preclosed (resp. semi-closed,α -closed, semi-preclosed) sets of (X, τ)containing A. It is known that pcl(A) (resp.scl(A), α cl(A), spcl(A)) is a preclosed (resp.semi-closed, α -closed, semi-preclosed) set. Forany subset A of an arbitrarily chosen topologicalspace, the semi-interior [5] (resp. α -interior[12], preinterior [13], semi-pre-interior [1]) of A,denoted by sint(A) (resp. α int(A), pint(A),
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PAKISTAN ACADEMY OF SCIENCESFounded
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Page 73: 136ACKNOWLEDGEMENTS: In a brief sta
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