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Acta Scientiarum. Technology ISSN: 1806-2563 [email protected] Universidade Estadual de Maringá Brasil Mishra, Sanjay On a- t-disconnectedness and - -connectedness in topological spaces Acta Scientiarum. Technology, vol. 37, núm. 3, julio-septiembre, 2015, pp. 395-399 Universidade Estadual de Maringá Maringá, Brasil Available in: http://www.redalyc.org/articulo.oa?id=303241163011 How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative Acta Scientiarum http://www.uem.br/acta ISSN printed: 1806-2563 ISSN on-line: 1807-8664 Doi: 10.4025/actascitechnol.v37i3.17332 On α- τ-disconnectedness and α- τ-connectedness in topological spaces Sanjay Mishra Department of Mathematics, School of Physical Sciences, Lovely Professional University, Punjab, India. E-mail: [email protected] ABSTRACT. The aims of this paper is to introduce new approach of separate sets, disconnected sets and connected sets called α- τ-separate sets, α- τ-disconnected sets and α- τ-connected sets of topological spaces with the help of α-open and α-closed sets. On the basis of new introduce approach, some relationship of ατ-disconnected and α- τ-connected set with α- τ-separate sets have been investigated thoroughly. Keywords: α-open set, α-closed set, α-closure, α- τ-separate sets, α- τ-disconnected sets and α- τ-connected sets. Sobre a α- τ-des-conectividade e a α- τ-conectividade em espaços topológicos RESUMO. Introduz-se nesse ensaio uma nova abordagem de conjuntos separados, conjuntos desconectados e conjuntos conectados chamados α- τ-conjuntos separados, α- τ-conjuntos desconectados e α- τ-conjuntos conectados de espaços topológicos, com o auxílio de conjuntos α- τ-aberto e α- τ-fechado. Baseados nessas abordagens, os relacionamentos de α- τ-conjunto desconectado e α- τ-conjunto conectado com α- τ-conjuntos separados foram analisados. Palavras-chave: α-conjunto aberto, α-conjunto fechado, α-fechamento, α- τ-conjuntos separados, α- τ-conjuntos desconectados, α- τ-conjuntos conectados. Introduction There are several natural approaches that can take to rigorously the concepts of connectedness for a topological spaces. Two most common approaches are connected and path connected and these concepts are applicable Intermediate Mean Value Theorem and use to help distinguish topological spaces. These concepts play a significant role in application in geographic information system studied by Egenhofer and Franzosa (1991), topological modelling studied by Clementini et al. (1994) and motion planning in robotics studied by Farber et al. (2003). The generalization of open and closed set as like open and closed sets was introduced by Njastad (1965) which is nearly to open and closed set respectively. These notion are plays significant role in general topology. In this paper, the new approaches of separate sets, disconnected sets and connected sets called separate set, disconnected sets and connected set with the help of open and closed set are firstly introduced. Further, some relationship disconnected concerning and connected sets with separate sets are also investigated. Acta Scientiarum. Technology Throughout this paper ( X , ) and ( X , ) will always be topological spaces. For a subset A of topological space X, Int ( A), Cl ( A), Cl ( A) and Int ( A) denote the interior, closure, closure and interior of A respectively and G is the open set for topology on X. Preliminaries We shall requires the following definitions and results. Definition 2.1. Levine (1963), defined a subset A of ( X , ) is semi-open if A Cl ( Int ( A)) and its complement is called semi-closed set. The family SO( X , ) of semi-open sets is not a topology on X. Definition 2.2. Mashhour et al. (1982), defined a subset A of ( X , ) is called pre-open locally dense or nearly open if A Int (Cl ( A)) and its complement is called pre-closed set. Theorem 2.3. According to Mashhour et al. (1982), the family PO( X , ) of pre-open sets is not a topology on X. Definition 2.4. Maheshwari and Jain (1982), defined a subset A of ( X , ) is called open if Maringá, v. 37, n. 3, p. 395-399, July-Sept., 2015 396 Mishra A Int (Cl ( Int ( A))) . The family of open sets of ( X , ) is a topology on X which is finer than and the complement of an open set is called an closed set. Theorem 2.5. According to Njastad (1965), for a subset A of ( X , ) , thee following are equivalent: A G and A H . ( A G ) ( A H ) . ( A G ) ( A H ) A . ( X , ) is said to be disconnected if there exists non-empty G and H in such that G H and G H X . Definition 3.4. Let ( X , ) be a topological space A ; A is semi open and pre-open; A B SO( X , ) , for all B in So( X , ) ; and A be non-empty subset of X. let G be A O / N , where O and N is nowhere arbitrary dense; There exists U contained in {G A : G } is a topology on A, called such that U A scl (U ) Int (Cl ( A)). Theorem 2.6. The intersection of semi-open (resp. pre-open) set and an open set is semiopen (resp. pre-open). Theorem 2.7. According to Njastad (1965), SO( X , ) SO( X , ) and PO( X , ) PO( X , ) . Theorem 2.8. According to Njastad (1965), the open sets A and B are disjoint if and only if Int(Cl ( Int ( A))) and Int (Cl ( Int ( B))) are disjoint. Definition 2.9. A point x in X is called an interior point of a set A in X if there exists A G such that y G with y x , i.e. x interior point of a set A in X if foe every G with x G and G A . Collection of all interior points of A is called interior of A which is denoted by Int ( A) . in X is called an Alternatively, we can define as Int ( A) by Int ( A) {G : G A} . Main Results Definition 3.1. Let ( X , ) and ( X , ) be topological spaces. Then the subsets A and B of ( X , ) are said to be separate sets if and only if A and B are non-empty set. A Cl (B) and B Cl ( A) are non-empty. Remark 3.2 If A and B are separate sets, then both of them are also disjoint sets. Definition 3.3. Let ( X , ) and ( X , ) be topological spaces. Then the subsets A of X is said disconnected, if there exists G and H in such that Acta Scientiarum. Technology in , then collection A the subspace or relative topology of topology . Theorem 3.5. If ( X , ) a disconnected space and ( X , ) is a topological space, then ( X , ) is disconnected. Proof. As ( X , ) disconnected and is finer than, hence by Theorem 3.1 ( X , ) is disconnected. Theorem 3.6. Let ( X , ) and ( X , ) are spaces, then ( X , ) is disconnected if and only if there exists non-empty proper subset of X which is both open and closed. Proof. Necessity: Let ( X , ) be disconnected. Then, by Definition 3.3 there exist non-empty sets G and H in such that G H is non-empty and G H X . Since H is open in show that G X H , but it is -closed. Hence G is non-empty proper subset of X which is -closed G H and as well -open. Sufficiency: Suppose A is non-empty proper subset of X such that it is -open as well closed. Now A is nonempty -closed show that X A is non-empty and -open. Suppose B X A , then A B X and A B . Thus A and B are non-empty disjoint -open as well as -closed subset of X such that A B X . Consequently X is -disconnected. Theorem 3.7. Every ( X , ) discrete space is -disconnected if the space contains more than one element. Proof. Let ( X , ) be discrete space such that X {a, b} contains more than one element. But is discrete topology so { , X ,{a},{b}} and family of all -open sets is { , X , {a}, {b}} . Maringá, v. 37, n. 3, p. 395-399, July-Sept., 2015 On α- τ-disconnectedness and α- τ-connectedness in topological spaces -closed sets are , X ,{a},{b} . Since {a} is non-empty proper subset of X which is both -open and -closed in X . Finally, we can All say that ( X , ) is -disconnected by Theorem 3.6. Theorem 3.8. A topological space ( X , ) is -connected 397 Sufficiency: disconnected Suppose and that M A NA A A - A is is a disconnection on A . By definition, we can say that M A , NA ; M A , NA A ; if and only if one non-empty subset which is both -open and -closed in X is X itself. Proof. Necessity: Assume that ( X , ) is - ( A M A ) ( A NA ) ; connected topological space. So, our assumption show that ( X , ) is not -disconnected i.e. there M A A M 1 , NA A N1 . But by (1) we can does not exist a pair of non-empty disjoin -open and -closed A and B such that, A B = X . This shows that there exist non-empty subsets (other than X ) which are both and -open and closed in X . Sufficiency: Suppose that ( X , ) is topological space such that the only non-empty subsets of X which is -open as well as -closed in X is X itself. By hypothesis, there does not exist a partition of the space X . Hence X is not disconnected i.e. connected. Theorem 3.9. Let A be a non-empty subset of A be the relative -disconnected if topological space ( X , ) . Let topology on A , then A is A is A -disconnected. Proof. Necessity: Let A be a -disconnected and let G H be a -disconnection on A . By Definition 3.3, there exists non-empty G and H in such that A G , A H ; ( A G ) ( A H ) ; ( A G ) ( A H ) A. and only if Now by the definition of relative topology, if G and H in ,then there exist G1 and H 1 in such that G A G and H A H . A Now by (1) 1 1 G1 and H 1 are non-empty. Hence A G1 and A H 1 are non-empty. Similarly by (2) and (3), we can say that ( A G1 ) ( A H1 ) and ( A G1 ) ( A H 1 ) A respectively. Consequently disconnected. Acta Scientiarum. Technology A is A - ( A M A ) ( A NA ) A . Now (2) there exists M 1 , N1 such that say A M 1 , A N1 . that Now ( A M ) A ( A M ) ( A A) M A M 1 . A 1 1 Similarly we can say that A NA A N1 . Now (3) ( A M 1 ) ( A N1 ) . ( A M 1 ) ( A N1 ) A . Finally, we can say that A is disconnected. Similarly (4) Theorem 3.10. The union of two non-empty -separate subsets of topological space ( X , ) is -disconnected. Proof. Let A and B be separate subsets of ( X , ) . Then by definition 3.1, we can say that A and B non-empty. A Cl ( B), B Cl ( A) ; A B . Let X Cl ( A) G and X Cl ( B) H . Then Cl (A) and Cl (B) are non-empty -closed subsets of X which shows that G and H are non-empty -open subsets of X . Since, G H ( X Cl ( A)) ( X Cl ( B )) X Cl ( A) Cl ( B ) we have ( A B) G ( A B) ( X Cl ( B)) [ A ( X Cl ( B))] [ B ( X Cl ( B))] B ( A B) G B . Similarly, ( A B) H A. Now (1) shows that ( A B) G , ( A B) H . Additionally, (3) shows that Maringá, v. 37, n. 3, p. 395-399, July-Sept., 2015 398 Mishra [( A B) H ] [( A B) G ] and Theorem 3.12. A subset Y of a topological space X is -disconnected if and only if it is union of two -separate sets. [( A B) H ] [( A B) G ] A B Proof. Necessity: Suppose Y A B, Finally, we can say that there exists G and H in such that ( A B) G , ( A B) H [( A B) H ] [( A B) G ] and [( A B) H ] [( A B) G ] A B . H is a -disconnection of A B. Hence A B is -disconnected. Theorem 3.11. Let ( X , ) and ( X , ) be topological spaces and A be a subset of X and let G H be a -disconnection of A. Then A G and A H are -separate subsets So, G of topological space ( X , ) . G H be a given disconnection of subset A of ( X , ) . To prove A G and A H are -separate subsets, Proof. Let we must show that A G and A H are non-empty; [Cl ( A G )] ( A H ) and [Cl ( A H )] ( A G ) ; ( A G ) and ( A H ) is non-empty; ( A G ) ( A H ) ; ( A G ) ( A H ) A. To prove (2) suppose it is not possible i.e. x Cl ( A G ) and x A, x H , that is ( A G ) H . Therefore ( A G ) ( A H ) . But it is contrary to (5). Finally our assumption that is wrong. Acta Scientiarum. Technology (Y G ) (Y H ) Y . Since (Y G ) and (Y H ) are separated sets, if we write A (Y G ) and B (Y H ), then by (3) Y A B . Finally, we can say that there exist two -separate sets A and B in X such that Y A B. Theorem 3.13. If Y is an -connected subset of topological space X such that Y A B, where A and B is -connected, then Proof. Since the inclusion Y A B holds by the hypothesis we have ( A B) Y Y which yields that Y (Y A) (Y B). Now we want to prove that (Y A), (Y B) . Suppose, Now, (Y A), (Y B) . (Y A) Cl (Y B)) . Then, there exists x Cl ( A G ) ( A H ) that (Y G ) (Y H ) ; (Y Cl (Y )) ( A Cl ( B)) (Y ( A Cl ( B)) Y Cl ( A G ) ( A H ) . implies Y G and Y H are non-empty; (Y A) Cl (Y B) (Y A) (Cl (Y ) Cl ( B)) Evidently, (4) (1). which A and B are -separate sets of X . A B is By theorem 3.10, disconnected. Hence, Y is -disconnected. Sufficiency: Let Y is -disconnected. To prove that there exists two -separate subsets of A, B in X such that Y A B. By assumption, Y is -disconnected show that there exists a -disconnection G H of Y . Therefore by Definition 3.3, we can say that there exists G H in such that Y A and Y B. Now by our assumption and definition, we can say that there exist G , H such that where i.e., Similarly, we can prove that Cl (Y A) (Y B) . Hence, from the above result we can say that Y is a union of two separate sets (Y A) and (Y B) . Consequently, Y is -disconnected. But this contradicts the fact that Y is -connected. Hence we can say that Now if (Y A) , (Y B) . Maringá, v. 37, n. 3, p. 395-399, July-Sept., 2015 On α- τ-disconnectedness and α- τ-connectedness in topological spaces (Y A) , then Y (Y B) (Y B) which gives that Y B. Similarly, we can prove that Y A if (Y A) . Conclusion We have introduced new approach of separate sets, disconnected sets and connected sets called separate sets, disconnected sets and connected sets of topological spaces with the help of open and closed sets and investigated their properties. The results of this paper will help to study various weak and strong form of connectedness and disconnectedness in topological spaces and it can be also applied for the study of topology in robotics, topological modeling and geographical information system. Acknowledgements I am very thankful to referee and my colleague Dr. A. K. Srivastva for providing useful commnets for improvement of the paper. 399 EGENHOFER, M. J.; FRANZOSA, R. D. Point-Set Topological Spatial Relations. International Journal of Geographical Information System, v. 5, n. 2, p. 161174, 1991. FARBER, M.; TABACHNIKOV, S.; YUZVINSKY, S. Topological robotics: Motion planning in projective spaces. International Mathematical Research Notices, v. 34, p. 1853-1870, 2003. LEVINE, N. Semi-open sets and Semi-continuity in Topological Spaces. The American Mathematical Monthly, v. 70, n. 1, p. 63-41, 1963. MASHHOUR, A. S.; MONSEF, A. E.; DEEB, S. N. E. On Precontinuous and Weak Precontinuous Mapping. Proceedings of the Mathematical and Physical Society of Egypt, v. 53, p. 47-53, 1982. MAHESHWARI, S. N.; JAIN, P. C. Some New Mappings. Mathematica, v. 24, n. 1-2, p. 53-55, 1982. NJASTAD, O. On Some Classes of Nearly Open Sets. Pacific Journal of Mathematics, v. 15, n. 3, p. 961-970, 1965. Reference Received on May 23, 2012. Accepted on September 28, 2012. CLEMENTINI, E.; SHARMA, J.; EGENHOFER, M. J. Modelling topological spatial relations Strategies for query processing. Computers and Graphics, v. 18, n. 6, p. 815-822, 1994. License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Acta Scientiarum. Technology Maringá, v. 37, n. 3, p. 395-399, July-Sept., 2015
