Abstracts in this research paper we are introducing the concept of m closed set and mt space,s discussed their properties, relation with other spaces and functions. In 1970, levine 14 initiated the study of socalled generalized closed sets. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. This new class falls strictly between the class of closed sets and rg closed sets. The converse of the above theorem need not be true, as seen from the following example. The proof follows from the definitions and the fact that closed sets are. Chinthamani vasthirani, 2assosiate professor, department of mathematics, sree sevugan annamalai college. Regular weakly closed sets, continuous maps and homeomorphisms in ideal topological spaces 2. In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. Since every closed set in a topological space is pre closed therefore pcla.
Abstract in this paper we introduce a new class of sets namely, gs closed sets, properties of this set are investigated and we. Regular open sets have been introduced and investigated by stone 17. This new class is defined over an initial universe and with a fixed set of parameters. A variety of topologies can be placed on a set to form a topological space.
Syed ali fathima department of mathematics sadakathullah appa college tirunelveli 627 011, india. The concepts of irwcontinuity, almost irwcontinuous, irwhomeomorphisms and related concepts in ideal topological spaces have been introduced and studied. A set is a set a which is equal to its kernel saturated set, i. The main aim of this paper is to introduce some new related closed sets. Any research work should result in addition to the existing knowledge of a particular concept. Suppose that fis continuous and let a y be a closed set. Metricandtopologicalspaces university of cambridge. Bcopen subsets of a topological space is denoted by. Introduction when we consider properties of a reasonable function, probably the. An important theorem about the product topology is tychonoffs theorem. A basis b for a topological space x is a set of open sets. In the present paper, we introduce and study a new class of sets called a. College of arts and science autonomous, pudukkottai 6 22 422, tamilnadu, india.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Since ynais open, f 1yna is open and therefore f 1a xnf 1yna is closed. We also introduce ggclosure, gginterior, ggneighbourhood, gglimit points. Also, as application using properties of rw open sets and rw closed stes. Throughout this paper x, represent nonempty topological spaces. In this paper, a new class of sets called regular wclosed briefly rwclosed sets in topological spaces is introduced and studied. Namely, we will discuss metric spaces, open sets, and closed sets. In this paper, aspects of generalized continuity and generalized closedness are explored. Benchalli and wali3 introduced the concept of rw closed sets in topological spaces.
The is defined and prove that it forms a topology on x. This concept was found to be useful and many results in general topology were improved. In this paper we introduce rw closed map from a topological space x to a topological space y as the image. Benchalli and wali3 introduced the concept of rwclosed sets in topological spaces. In this research paper, a new class of open sets called ggopen sets in topological space are introduced and studied. Rwclosed sets, rwopen sets, locally closed sets, rwlocally closed sets 1. The notion of generalized closed sets in ideal topological spaces was studied by dontchev et. A space x is said to be fuzzy rwt2 if for each pair of distinct points. In this paper, the concept of regular w closed rw closed sets in topological spaces introduced in 1 is further studied.
In this research paper, a new class of continuous functions called ggcontinuous maps in topological space are introduced and studied. Pdf on rwclosed sets in bitopological spaces researchgate. The class of all g sclosed set s in a topological space x. The concept of t 12 gts depends in turn on the concept of a generalized closed set. Introduction to topological spaces and setvalued maps. Also we discuss some of their properties and investigate the relations between the associated. A proof that relies only on the existence of certain open sets will also hold for any finer topology, and. Fuzzy almost contra rwcontinuous functions in topological spaces 1 intersection of fuzzy regular closed sets. A base for an lfuzzy topology f on a set x is a collection 93 c y such that, for each u e r there exists gu c g. Applications of closed sets in intuitionistic fuzzy topological space p. The aim of this paper is to introduce and study a new class of generalized closed sets called gp closed sets in topological spaces using gp closed sets. 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. Hadi 3 1,2 department of mathematics, college of education for pure science almuthanna university, samawah, iraq.
Applications of closed sets in intuitionistic fuzzy. Wali 9 introduce rw continuous, rw homeomorphism using rw closed sets in topologcal spaces. Pdf on ggcontinuous and ggirresolute maps in topological. In this paper a new class of soft sets called soft regular weakly closed sets briefly srwclosed sets in soft topological spaces is introduced and studied. We also introduce ggirresolute maps, strongly ggcontinuous. Rajarubi abstract in this paper, we introduce a new class of sets called. 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 introduce and study a new class of sets called. In this research paper, a new type of closed sets called generalization of generalized closed sets ggclosed sets are introduced and studied in a topological space x. The property we want to maintain in a topological space is that of nearness. The main aim of this paper is to introduce some new related closed sets in the same space and study. On generalized pre regular weakly gprwclosed sets in. We will allow shapes to be changed, but without tearing them.
In this paper, using the concept of fuzzy rw open sets. On generalized closed sets and generalized preclosed sets. A sub set a of a topological space x is said to be rw open sets, if u intcla,whenever u a and u is rw closed sets in x. X, it is proved that the complement of grwinterior of a is the grwclosure of the complement.
Sundaram and others published on w closed sets in topology find, read and cite all the research you need on researchgate. This new class of sets lies between the class of regular weakly closed briefly, rwclosed sets and the class of. We explore the relationship of generalized closed sets with several separation axioms, t 0, t 1, t 12, regularity, and normality 32, 33. Pdf in this paper, a new class of sets called regular wclosed briefly rwclosed sets in topological spaces is introduced and studied. Further closed sets like i rg,i rw were further developed by navaneethakrishnan 10 and a. On regular generalized open sets in topological space citeseerx. Parimelazhagan2 1research scholar,karpagam university,india 2department of science and humanities, karpagam college of engineering, anna university, india abstract. Such an effort not only widens the scope of the concept but also encourages others to explore new and newer ideas. So, one can imagine that, how important the concept of closed sets is in the topological spaces. Pdf on generalized pre regular weakly gprwclosed sets.
Any product of closed subsets of x i is a closed set in x. The open sets in a topological space are those sets a for which a0. Also the concept of grwclosure in topological spaces using the notions of grwclosed sets, and some related results will be obtained. Pushpalatha karpagam college of engineering, coimbatore, india government arts college, udumalpet, india abstract. Throughout this paper x, and y, or simply x and y represents the nonempty topological spaces on which no separation axiom are assumed, unless. The aim of this paper is to introduce a new class of sets namely rg closed sets in topological spaces.
Generalized pre regular weakly closed sets 1985 proof. Topology underlies all of analysis, and especially certain large spaces such as the dual of l1z. Also some of their properties have been investigated. Maki 12 introduced the notion of sets in topological spaces. Introduction kuratowski and sierpinski 11 introduced the notion of locally closed sets in topological 6, a subset of a topological space x. A topological space is the most basic concept of a set. This class was obtained by generalizing closed sets via rg open sets which was introduced by n. First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological spaces. In this paper, we study some of the properties of interval valued fuzzy rw closed and interval valued fuzzy rw open sets in interval valued fuzzy topological spaces and prove some results on these.
In this paper, we introduce the concepts of i rw homeomorphism and study the relationship with ihomeomorphisms. In this paper the structure of these sets and classes of sets are investigated, and some applications are given. In a complete metric space, a closed set is a set which is closed under the limit operation. In this paper, a new class of sets called regular w closed briefly rwclosed sets in topological spaces is introduced and studied. In a topological space, a closed set can be defined as a set which contains all its limit points. Fuzzyalmost contra rwcontinuous functions in topological spaces. In this paper a class of sets called g closed sets and g open sets and a class of maps in topological spaces is introduced and some of its properties are discussed.
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. Kuratowski 10 introduced the concept of ideal topological spaces. Pdf on rwclosed sets in topological spaces researchgate. Preliminaries in this section we recollect the following basic definitions which are used in this paper. Some basic properties of this new class of soft sets are investigated. The standard material on the notions of gopen, gopen sets and some definitions and results that are needed are presented first. Bagyalakshmi assistant professor, assistant professor, department of mathematics department of mathematics, chikkanna government arts college ajk college of arts and science.
In particular, if one considers the space x r i of all real valued functions on i, convergence in the product topology is the same as pointwise convergence of functions. In this paper, the authors introduce and study the concept of a new class of closed sets called weakly generalized e. A new closure operator grwclosure in topological spaces. Chapter 2 regular weakly closed sets, continuous maps and. L, assistant professor, nirmala college for women, coimbatore, tamil nadu. Compactness in fuzzy topological spaces 549 and u v v and a9 are similarly defined.
It is assumed that measure theory and metric spaces are already known to the reader. A subset a of x is called a g sclosed set if scl a u whenever a u, u is gs open. Free topology books download ebooks online textbooks. On generalized closed sets in generalized topological spaces. Generalized closed sets and open sets in topological. Closed sets in nano topological spaces qays hatem imran 1, murtadha m. R, pg student, nirmala college for women, coimbatore,tamil nadu. Introduction the concept of generalized closed sets and generalization of closed sets in topological spaces was introduced by levine 7 in 1970.
Ideals in topological spaces were introduced by kuratowski 36 and vaidyanathaswamy 65. On regular generalized open sets in topological space. Finite spaces have canonical minimal bases, which we describe next. The notion of kuratowski operator plays a vital role in defining ideal topological. Jeyaraman, 1 assistant professor, department of mathematics, rdm government arts college, sivagangai. On nano generalized alpha generalized closed sets in nano. In this paper, a new class of sets called regular wclosed briefly rw closed sets in topological spaces is introduced and studied. In 1970, levine9 introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces.
Soft sets, soft topological spaces, w closed sets, sw closed sets, rw closed sets, srw closed sets. Srwclosed sets in soft topological spaces open access journals. On rwclosed sets in topological spaces rims, kyoto university. For a subset a of a topological space x, cla, inta, scla, cla and spcla denote the closure of a, interior of a, the. Andrijevic1 introduced semi preopen sets in general topology. Introduction the concept explored via ideals has a lengthy and interesting historic development. In this paper we define and study the properties of a new topological axioms called rw t. Since every closed set in a topological space is preclosed therefore pcla. On generalized closed sets in generalized topological. Studies on generalizations of generalized closed sets and maps in topological spaces, ph. Using generalized closed sets, dunham 1982 introduced the. Notes on interval valued fuzzy rwclosed, interval valued.