Acta Scientiarum. Technology ISSN: Universidade Estadual de Maringá Brasil

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1 cta Scientiarum Technology ISSN: Universidade Estadual de Maringá Brasil Mishra, Sanjay On a tdisconnectedness connectedness in topological spaces cta Scientiarum Technology, vol 37, núm 3, julioseptiembre, 205, pp Universidade Estadual de Maringá Maringá, Brasil vailable in: How to cite Complete issue More information about this article Journal's homepage in redalycorg Scientific Information System Network of Scientific Journals from Latin merica, the Caribbean, Spain Portugal Nonprofit academic project, developed under the open access initiative

2 cta Scientiarum ISSN printed: ISSN online: Doi: 04025/actascitechnolv37i37332 On α τdisconnectedness α τconnectedness in topological spaces Sanjay Mishra Department of Mathematics, School of Physical Sciences, Lovely Professional University, Punjab, India BSTRCT The aims of this paper is to introduce new approach of separate sets, disconnected sets connected sets called α τseparate sets, α τdisconnected sets α τconnected sets of topological spaces with the help of αopen αclosed sets On the basis of new introduce approach, some relationship of α τdisconnected α τconnected set with α τseparate sets have been investigated thoroughly Keywords: αopen set, αclosed set, αclosure, α τseparate sets, α τdisconnected sets α τconnected sets Sobre a α τdesconectividade e a α τconectividade em espaços topológicos Introduction RESUMO Introduzse nesse ensaio uma nova abordagem de conjuntos separados, conjuntos desconectados e conjuntos conectados chamados α τconjuntos separados, α τconjuntos desconectados e α τconjuntos conectados de espaços topológicos, com o auxílio de conjuntos α τaberto e α τfechado Baseados nessas abordagens, os relacionamentos de α τconjunto desconectado e α τconjunto conectado com α τconjuntos separados foram analisados Palavraschave: αconjunto aberto, αconjunto fechado, αfechamento, α τconjuntos separados, α τconjuntos desconectados, α τconjuntos conectados There are several natural approaches that can take to rigorously the concepts of connectedness for a topological spaces Two most common approaches are connected path connected these concepts are applicable Intermediate Mean Value Theorem use to help distinguish topological spaces These concepts play a significant role in application in geographic information system studied by Egenhofer Franzosa 99), topological modelling studied by Clementini et al 994) motion planning in robotics studied by Farber et al 2003) The generalization of open closed set as like open closed sets was introduced by Njastad 965) which is nearly to open closed set respectively These notion are plays significant role in general topology In this paper, the new approaches of separate sets, disconnected sets connected sets called separate set, disconnected sets connected set with the help of open closed set are firstly introduced Further, some relationship concerning disconnected connected sets with separate sets are also investigated Throughout this paper X, ) X, ) will always be topological spaces For a subset of topological space X, Int ), Cl ), Cl ) Int ) denote the interior, closure, closure interior of respectively G is the cta Scientiarum Technology Maringá, v 37, n 3, p , JulySept, 205 open set for topology on X Preliminaries We shall requires the following definitions results Definition 2 Levine 963), defined a subset of X, ) is semiopen if Cl Int )) its complement is called semiclosed set The family SO X, ) of semiopen sets is not a topology on X Definition 22 Mashhour et al 982), defined a subset of X, ) is called preopen locally dense or nearly open if Int Cl )) its complement is called preclosed set Theorem 23 ccording to Mashhour et al 982), the family PO X, ) of preopen sets is not a topology on X Definition 24 Maheshwari Jain 982), defined a subset of X, ) is called open if

3 396 Mishra Int Cl Int ))) The family of open sets of X, ) is a topology on X which is finer than the complement of an open set is called an closed set Theorem 25 ccording to Njastad 965), for a subset of X, ), thee following are equivalent: ; is semi open preopen; B SO X, ), for all B in So X, ) ; O / N, where O N is nowhere dense; There exists U contained in U scl U) Int Cl )) Theorem 26 The intersection of semiopen resp preopen) set an open set is semiopen resp preopen) Theorem 27 ccording to Njastad 965), SO X, ) SO X, ) PO X, ) PO X, ) Theorem 28 ccording to Njastad 965), the open sets B are disjoint if only if Int Cl Int ))) Int Cl Int )) are disjoint Definition 29 point x in X is called an interior point of a set in X if there exists G y G with y x, ie x in X is called an interior point of a set in X if foe every G with x G G Collection of all interior points of is called interior of which is denoted by Int ) lternatively, we can define as Int ) by Int ) { G : G } Main Results Definition 3 Let X, ) X, ) be topological spaces Then the subsets B of X, ) are said to be separate sets if only if B are nonempty set Cl B Cl ) are nonempty Remark 32 If B are separate sets, then both of them are also disjoint sets Definition 33 Let X, ) X, ) be topological spaces Then the subsets of X is said disconnected, if there exists G H in H G G ) H ) G ) H ) X, ) is said to be disconnected if there exists nonempty G H in such that G H G H X Definition 34 Let X, ) be a topological space be nonempty subset of X let G be arbitrary in, then collection { G : G } is a topology on, called the subspace or relative topology of topology Theorem 35 If X, ) a disconnected space X, ) is a topological space, then X, ) is disconnected Proof s X, ) disconnected is finer than, hence by Theorem 3 X, ) is disconnected Theorem 36 Let X, ) X, ) are spaces, then X, ) is disconnected if only if there exists nonempty proper subset of X which is both open closed Proof Necessity: Let X, ) be disconnected Then, by Definition 33 there exist nonempty sets G H in G is nonempty G H X Since H G H H is open in show thatg X H, but it is closed Hence G is nonempty proper subset of X which is closed as well open Sufficiency: Suppose is nonempty proper subset of X it is open as well closed Now is nonempty closed show that X is nonempty open Suppose B X, then B X B Thus B are nonempty disjoint open as well as closed subset of X B X Consequently X is disconnected Theorem 37 Every X, ) discrete space is disconnected if the space contains more than one element Proof Let X, ) be discrete space X { a, b} contains more than one element But is discrete topology so {, X,{ a},{ b}} family of all open sets is {, X,{ a},{ b}} cta Scientiarum Technology Maringá, v 37, n 3, p , JulySept, 205

4 On α τdisconnectedness α τconnectedness in topological spaces 397 ll closed sets are, X,{ a},{ b} Since {a} is nonempty proper subset of X which is both open closed in X Finally, we can say that X, ) is disconnected by Theorem 36 Theorem 38 topological space X, ) is connected if only if one nonempty subset which is both open closed in X is X itself Proof Necessity: ssume that X, ) is connected topological space So, our assumption show that X, ) is not disconnected ie there does not exist a pair of nonempty disjoin open closed B, B = X This shows that there exist nonempty subsets other than X ) which are both open closed in X Sufficiency: Suppose that X, ) is topological space the only nonempty 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 ie connected Theorem 39 Let be a nonempty subset of topological space X, ) Let be the relative topology on, then is disconnected if only if is disconnected Proof Necessity: Let be a disconnected let G H be a disconnection on By Definition 33, there exists nonempty G H in G, H ; G ) H ) ; G ) H ) Now by the definition of relative topology, if G H in,then there exist G H in G G H H Now by ) G H are nonempty Hence G H are nonempty Similarly by 2) 3), we can say that G ) H ) G ) H ) respectively Consequently is disconnected Sufficiency: Suppose that is disconnected M N is a disconnection on By definition, we can say that M M, ; N, N ; M ) N ) ; M ) N ) Now 2) there exists M, N M M, N N But by ) we can say that M, N Now M ) M ) ) M M Similarly we can say that N N Now 3) M ) N ) Similarly 4) M ) N ) Finally, we can say that is disconnected Theorem 30 The union of two nonempty separate subsets of topological space X, ) is disconnected Proof Let B be separate subsets of X, ) Then by definition 3, we can say that B nonempty Cl, B Cl ) ; B Let X Cl ) G X Cl H Then Cl ) Cl are nonempty closed subsets of X which shows that G H are nonempty open subsets of X Since, G H we have X Cl )) X Cl ) X Cl ) Cl G X Cl ) [ X Cl )] [ B X Cl )] B G B Similarly, H Now ) shows that G, H dditionally, 3) shows that cta Scientiarum Technology Maringá, v 37, n 3, p , JulySept, 205

5 398 Mishra [ H ] [ G ] [ H ] [ G ] B Finally, we can say that there exists H in G, H [ H ] [ G ] [ H ] [ G ] B G So, G H is a disconnection of B Hence B is disconnected Theorem 3 Let X, ) X, ) be topological spaces be a subset of X let G be a disconnection of Then H G H of topological space X, ) Proof Let H are separate subsets G be a given disconnection of subset of X, ) To prove G H are separate subsets, we must show that G H are nonempty; [ Cl G )] H ) [ Cl H )] G ) ; Now by our assumption definition, we can say that there exist G, H G ) H ) is nonempty; G ) H ) ; G ) H ) Evidently, 4) ) To prove 2) suppose it is not possible ie Cl G ) H ) Then, there exists x Cl G ) H ) which implies that x Cl G ) x, x H, that is G ) H Therefore G ) H ) But it is contrary to 5) Finally our assumption that is wrong Theorem 32 subset Y of a topological space X is disconnected if only if it is union of two separate sets Proof Necessity: Suppose Y B, where B are separate sets of X By theorem 30, B is disconnected Hence, Y is disconnected Sufficiency: Let Y is disconnected To prove that there exists two separate subsets of, B in X Y B By assumption, Y is disconnected show that there exists a disconnection G H of Y Therefore by Definition 33, we can say that there exists H in Y Y H are nonempty; G Y G ) Y H ) ; Y G ) Y H ) Y G Since Y G ) Y H ) are separated sets, if we write Y G ) B Y H ), then by 3) Y B Finally, we can say that there exist two separate sets B in X such that Y B Theorem 33 If Y is an connected subset of topological space X Y B, where B is connected, then Y Y B Proof Since the inclusion Y B holds by the hypothesis we have Y Y which yields that Y Y ) Y Now we want to prove that Y ), Y Suppose, Y ), Y Now, Y ) Cl Y Y ) Cl Y) Cl ) Y Cl Y)) Cl ) Y Cl ) Y Y ) Cl )) ie, Y B Similarly, we can prove that Cl Y ) Y Hence, from the above result we can say that Y is a union of two separate sets Y ) Y Consequently, Y is disconnected But this contradicts the fact that Y is connected Hence we can say that Y ), Y Now if cta Scientiarum Technology Maringá, v 37, n 3, p , JulySept, 205

6 On α τdisconnectedness α τconnectedness in topological spaces 399 Y ), then Y Y Y which gives that Y B Similarly, we can prove that Y if Y ) Conclusion We have introduced new approach of separate sets, disconnected sets connected sets called separate sets, disconnected sets connected sets of topological spaces with the help of open closed sets investigated their properties The results of this paper will help to study various weak strong form of connectedness disconnectedness in topological spaces it can be also applied for the study of topology in robotics, topological modeling geographical information system cknowledgements I am very thankful to referee my colleague Dr K Srivastva for providing useful commnets for improvement of the paper Reference CLEMENTINI, E; SHRM, J; EGENHOFER, M J Modelling topological spatial relations Strategies for query processing Computers Graphics, v 8, n 6, p 85822, 994 EGENHOFER, M J; FRNZOS, R D PointSet Topological Spatial Relations International Journal of Geographical Information System, v 5, n 2, p 6 74, 99 FRBER, M; TBCHNIKOV, S; YUZVINSKY, S Topological robotics: Motion planning in projective spaces International Mathematical Research Notices, v 34, p , 2003 LEVINE, N Semiopen sets Semicontinuity in Topological Spaces The merican Mathematical Monthly, v 70, n, p 634, 963 MSHHOUR, S; MONSEF, E; DEEB, S N E On Precontinuous Weak Precontinuous Mapping Proceedings of the Mathematical Physical Society of Egypt, v 53, p 4753, 982 MHESHWRI, S N; JIN, P C Some New Mappings Mathematica, v 24, n 2, p 5355, 982 NJSTD, O On Some Classes of Nearly Open Sets Pacific Journal of Mathematics, v 5, n 3, p 96970, 965 Received on May 23, 202 ccepted on September 28, 202 License information: This is an openaccess article distributed under the terms of the Creative Commons ttribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited cta Scientiarum Technology Maringá, v 37, n 3, p , JulySept, 205