December 2016 on generalized closed sets in generalized topological spaces b. Properties of these sets are investigated for topological spaces and generalized topological spaces. Second, we allow for the possibility that the whole space is not open. New class of generalized closed sets in supra topological. Although the following discussion is provided for generalized topological space, it is also valid for topological spaces as every topological space is a generalized topological space as well.
Pdf soft generalized closed sets in soft topological spaces. Several specific types of generalized sets of a generalized topological space have been defined and investigated for various purposes from. Neutrosophic set, generalized neutrosophic set, neutrosophic topology introduction and preliminaries. Closed sets are fundamental objects in a topologicalspace. Veera kumar 12 introduced g closed sets in topological spaces. In this paper we introduce and characterize some new generalized locally closed sets known as. Some new regular generalized closed sets in ideal topological.
Also we study some of its basic properties and investigate the relations between the associated topology. A subset a of x is said to be bgclosed if bcla u whenever a u and u is g. Parimelazhagan 7 defined closed sets of topological spaces. A subset a of x is said to be ideal generalized closed set briefly ig closed set if a u whenever a u and u is open. Maki 12 introduced the notion of sets in topological spaces. On soft generalized closed sets in a soft topological. Determine some properties of regularity and compare with other types of regular spaces. In 2002, a cs asz ar 3 introduced the concept of generalized topological space or simply gt space. Several specific types of generalized sets of a generalized topological space have been defined and investigated for various purposes from time to time in. In this paper generalized alpha closed sets and generalized alpha open sets are presented. Unification of generalized open sets on topological spaces.
The open and closed sets of a topological space examples 1 fold unfold. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms are. Generalized bold0mu mumu operations on fuzzy topological. A set is a set which is equal to its kernel saturated set, that is, to the intersection of all open supersets of. Rajarubi abstract in this paper, we introduce a new class of sets called. Chauhan2 1 department of mathematics, atmaram sanatan dharma college, university of delhi. Closed sets are fundamental objects in a topological space. Theory, relations with generalized algebraic structures and applications, volume 199 1st edition. The investigation on generalization of closed set has lead to significant contribution to the theory of separation axiom, generalization of continuity and irresolute functions. New generalized topologies on generalized topological. Conclusion we have introduced soft generalized closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters. Closed sets, hausdor spaces, and closure of a set 9 8. On pre open sets in topological spaces and its applications. Also, we introduce some characterizations and properties for these concepts.
Between closed sets and g closed sets springerlink. Introduction julian dontchev, maximilian ganster and takashi noiri 2000 has introduced the concept of open sets in topological spaces. Knebusch and their strictly continuous mappings begins. This notion had been studied extensively in the recent years by many topologists since generalized closed sets are not only the generalization of closed sets. Journal of mathematics and computing applications, vol. In this paper we have introduced a new class of sets called gsp closed sets which is properly placed in between the class of closed sets and gsp closed sets. Background in set theory, topology, connected spaces, compact spaces, metric spaces, normal spaces, algebraic topology and homotopy theory, categories and paths, path lifting and covering spaces, global topology. On preopen sets in topological spaces and its applications a. Generalized closed sets and open sets in topological. Namely, we will discuss metric spaces, open sets, and closed sets. Introduction in chapter i we looked at properties of sets, and in chapter ii we added some additional structure to a set a distance function to create a pseudomet. Evidently specifying the open subsets is equivalent to specfying the closed subsets.
Mathematics free fulltext on generalized closed sets and. Sheik john 24 veera kumar 26 introduced the notion of closed sets gclosed sets. In this paper another generalization of igclosed sets namely g i eclosed set is defined using semipre local function. In the present paper, we introduce soft generalized closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters. Further, gspcontinuous, and gsp irresolute mappings are also introduced and investigated. Murugesan3 1, 2assistant professor, department of mathematics, panimalar institute of technology, chennai, india 3associate professor, pg and research department of mathematics, sri s. Csaszar has developed a theory of generalized topological spaces 2, 3, 5, 4. Anbarasi rodrigo2 research scholar pg and research department of mathematics voc college, thoothukudi, india. A new class of generalized closed sets in topological spaces 335 theorem 3. As an application, we introduce two news paces namely, tgspspace, gtgspspace. The authors introduce new type of closed set named gpclosed set in the topological. On generalized closed sets and generalized pre closed sets in neutrosophic topological spaces. Finally, we proved that the our new normality properties are preserved under some types of continuous functions between bitopological spaces. Informally, 3 and 4 say, respectively, that cis closed under.
Also, it is free of the difficulties present in the theories of fuzzy sets. Generalized closed sets in ideal topological spaces. On generalized closed sets in generalized topological spaces. Antony rex rodgio and et al2 introduce d closed sets in topological spaces and studied their properties. Semiconnectedness is characterized by using regular sets.
Elatik department of mathematics, faculty of science, tanat university, tanta, egypt abstract in this paper, we consider the class of preopen sets in topological spaces and investigate some of their properties. Using generalized closed sets, dunham 1982 introduced the concept of generalized closure operator cl and. In this paper, the concept of generalized neutrosophic preclosed sets and generalized neutrosophic. We also call aendowed with the subspace topology a subspace of x. A set is a set a which is equal to its kernel saturated set, i. In this paper, the authors introduce and study the concept of a new class of closed sets called weakly generalized e. A new class of closed sets in strong generalized topological spaces g. The concept of t 12 gts depends in turn on the concept of a generalized closed set. After given the fundamental definitions of generalized neutrosophic set operations, we obtain several properties, and discussed the relationship between generalized neutrosophic sets and others. A family of soft sets has the nite intersection property fip if the intersection of the members of each nite subfamily of is not a null soft set. In the present paper we introduce soft topological spaces which are defined over an initial universe with a fixed set of parameters.
Pdf in this paper we study generalized closed sets in the sense of n. T h e n t h e f o l l o w i n gs t a t e m e n t sh o l d. The concept of generalized closed sets and generalized open sets was first. We also investigate several properties of such sets. Ig closed sets, semi i closed set, pre i closed set. The subspace topology on ais given by the collection fa\ujuopen in xg thus a subset v ais open in this topology if and only if there exists an open subset u x such that v a\u. The aim of this work is to introduce operations on fuzzy topological spaces and to use them to study fuzzy generalized closed sets and fuzzy generalized open sets. A new class of generalized closed sets in topological spaces. Finally we show that certain results of several publications on the concepts of weakness and strength of fuzzy generalized closed sets are.
First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological spaces. Between closed sets and closed sets in topological spaces. Some new regular generalized closed sets in ideal topological spaces umit karab. Recently, on soft sets, soft topological space has. This paper committed to the investigation of neutrosophic topological spaces. In this paper, we introduce a new type of closed sets in bitopological space x. Moreover, the union and intersection of two soft g closed sets. For any set x, we have a boolean algebra px of subsets of x, so this algebra of sets may be treated as a small category with inclusions as morphisms. Levine4 introduced the notion of generalized closed brie y g closed sets. The purpose of this paper is to introduce and study the concept of.
The closed subsets of a topological space satsify the following properties. Paper 2, section i 4e metric and topological spaces. Parimelazhagan2 1research scholar,karpagam university,india 2department of science and humanities, karpagam college of engineering, anna university, india abstract. In mathematics, a metric space is a set together with a metric on the set.
We explore the relationship of generalized closed sets with several separation axioms, t 0, t 1, t 12, regularity, and normality 32, 33. The open and closed sets of a topological space examples 1. And some basic properties of this new class are investigated by using the concept of weak structure. Using generalized closed sets, dunham 1982 introduced the concept of generalized. A sufficient condition for a soft g closed set to be a soft closed is also presented. We also use this notion to consider new weak form of continuities with these sets. In 1997, fuzzy generalized closed set briefly was introduced by g. Further it covers metric spaces, continuity and open sets for metric spaces, closed sets for metric spaces, topological spaces, interior and closure, more on topological structures, hausdorff spaces and compactness. On generalized topological spaces semantic scholar. In 1986, maki continued the work of levine and dunham on generalized closed sets and closure operators by introducing the notion of sets in topological spaces. Applying these sets, we obtain a new space which is called t 34space. Hereditarily homogeneous generalized topological spaces. We then looked at some of the most basic definitions and properties of pseudometric spaces.
If x is a topological space and x 2 x, show that there is a connected subspace k x of x so that if s is any other connected subspace containing x then s k x. In 1937, regular open sets were introduced and used to define the semiregularization space of a topological space. The aim of this paper is to investigate the properties of these sets in the ideal topological spaces. On a generalized closed sets in ideal topological spaces. If you define the topology with open sets then a closed set is a complement of a member of the topology. There are many other equivalent ways to define a topological space. As an application, we introduce two news paces namely, tgsp space, gtgsp space.
Ti i 0, 1 2,1,2 spaces are characterized them using. New types of generalized closed sets in bitopological spaces. It is then applied in reconstructing the important elements of. Pious missier1 associate professor of mathematics pg and research department of mathematics voc college, thoothukudi, india. Quite recently, by using semiopen respopen, preopen. Many researchers 1, 2, 4, 6, 18 turned their attention to define new sets and functions in topological spaces.
Generalized topological spaces in the sense of csaszar have two main features which distinguish them from typical topologies. A subset a of a topological space x,t is called a semi generalized. The investigation on generalized of closed set has lead to signi cant contribution to the theory of separation axiom, generalization of continuity and covering properties. Decompositions of regular open sets and regular closed sets are provided using regular sets.
Further, we introduce the notions of interior, closure, boundary, exterior and study some of their properties. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms are introduced and their basic properties are investigated. Many variations of closed sets were introduced and investigated. By definition of a subset a of a topological space x is called generalized closed briefly, g closed set if cla. In this paper we introduce definitions of generalized neutrosophic sets. On generalized topological spaces arturpiekosz abstract in this paper a systematic study of the category gtsof generalized topological spaces in the sense of h. This paper covers some recent progress in the study of sgopen sets, sgcompact spaces, nscattered spaces and some related concepts. In 1970, levine 16 initiated the study of socalled generalized closed sets. Generalized alpha closed sets in neutrosophic topological. Alternative characterizations of topological spaces closed. In 1980, levine and dunham 7 further characterized some more properties of generalized closed sets. A metric on a space induces topological properties like open and closed sets, which lead to.
A subset a of x is said to be bg closed if bcla u whenever a u and u is g. On some locally closed sets and spaces in ideal topological spaces. First, these families of subsets are not closed under intersections. The authors introduce new type of closed set named gp closed set in the topological. Pdf closed sets in topological spaces researchgate. In 1970, levine9 introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. On characterizations of nano rgbclosed sets in nano. In this work, we introduce the notion of generalized. Free topology books download ebooks online textbooks. Ais a family of sets in cindexed by some index set a,then a o c. Using generalized closed sets, dunham8 introduced the concept of the closure operator cl and a new topology and studied some of their properties. Topological properties defined in terms of generalized.