Every topological space x is an open dense subspace of a compact space having at most one point more than x, by the alexandroff onepoint compactification. Neutrosophic topological spaces are introduced by salama and alblowi9. The soft topological spaces and some their related concepts have been studied by shabir and naz in 1. An introduction to fuzzy soft topological spaces abdulk adir ayguno glu vildan c. Firstly, we recall some basic properties of fuzzy soft. In this work, we define the soft topology on a soft set, and present its related properties. Furthermore, the concept of soft continuous and almost soft continuous functions are defined by using the soft open sets. The main purpose of this paper is to introduce and study new classes of soft closed sets like soft regular generalized bclosed sets in soft topological. We characterize soft compact space and study their basic properties. Soft regular generalized closed sets in soft topological spaces. Informally, 3 and 4 say, respectively, that cis closed under.
We explore some basic properties of these concepts. Topological spaces in this section, we introduce the concept of g closed sets in topological spaces and study some of its properties. Then we discuss some basic properties of soft topological spaces and define soft open and closed sets. Some results on soft element and soft topological space. Soft set theory is a newly emerging tool to deal with uncertain problems and has been studied by researchers in theory and practice. Mathur, 22 introduced nearly compact spaces in topological spaces.
In particular, we investigate the soft regular spaces and some properties of them. Fuzzy soft generalized closed sets in fuzzy soft topological spaces. We defined quasiopen fuzzy soft set over fuzzy topological space and we study the properties of extended union of two quasiopen fuzzy soft set over fuzzy topological space. Third is to define soft compactness and generalize alexander subbase theorem and tychonoff theorem to the soft topological spaces. Revised 29 july 2011 accepted 2 august 2011 abstract. Suppose h is a subset of x such that f h is closed where h denotes the closure of h. Abstract the purpose of this paper is to form some new. Pdf soft generalized closed sets in soft topological spaces.
Finally, we prove that the category of fuzzy soft topological spaces fstop is a topological category over set. Paper 1, section ii 12e metric and topological spaces. Compactness in fuzzy topological spaces 549 and u v v and a9 are similarly defined. All over the globe, soft set theory is a topic of interest for many authors working in diverse areas because of its rich potential for applications in. Fuzzy soft set over a fuzzy topological space springerlink. Soft regular generalized closed sets in soft topological.
In section 6 we give some questions on soft topological spaces. Gn 29 may 2019 soft n topological spaces giorgio nordo mift dipartimento di scienze matematiche e informatiche, scienze fisiche e scienze della terra. A soft topological space xe is said to be soft compact space if each soft open cover of x has a nite subcover. In order to study a few re sults of fuzzy soft topological spaces properly, roy and samanta 6 restated a fe w basic definitions of fuzzy soft sets in the following manner definition 2. Some results on soft element and soft topological space polat. Chen, b soft semiopen sets and related properties in soft topological spaces. They also studied some of basic concepts of soft topological spaces. On some generated soft topological spaces and soft homogeneity.
A fuzzy soft set g c in fsf e is said to be a neighborhood of a fuzzy soft set h a if and only if there exists a fuzzy soft open set k b such that h a. The first aim of this study is to define soft topological spaces and to define soft continuity of soft mappings. We also introduce the concepts of soft interior point, soft interior. The purpose of this paper is to study soft sets and soft topological spaces. On soft compact soft generalized topological spaces. Pdf remarks on soft topological spaces researchgate. This concept is discussed with a view to find new soft topologies from the original one, called. The definitions of subspace soft generalized topology and soft continuity of soft functions are introduced. In this paper a new class of soft sets called soft gclosed sets in soft topological spaces is introduced and studied.
Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. A fuzzy soft subspace of a fuzzy soft t 0 space is fuzzy soft t 0, i. But we found that some results are incorrect see their remark 3. In section 5 we introduce and study notions concerning the soft. Buvaneswari department of mathematics, alagappa govt. In this paper we introduce fuzzy soft bopen set in fuzzy soft topological spaces.
Fuzzy neutrosophic topological spaces and fuzzy neutrosophic soft topological spaces are introduced by arockiarani, and martinajency1,2. Some properties of soft compact spaces in soft topological spaces in this section, the concept of soft space is introduced and studied the concept of soft compactness in terms of soft compact spaces. We introduce the notion of fuzzy soft set over common universe of fuzzy topological space. Soft set, soft generalized topological space, soft generalized. The concept of soft sets is introduced as a general mathematical tool for dealing with uncertainty. Shabir and naz introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. Some notes on soft topological spaces springerlink. We anticipate that the results in this paper can be stimulated to the further study on soft topology to accomplish genenral framework. Ais a family of sets in cindexed by some index set a,then a o c. Moreover we introduce fuzzy soft bclosure and fuzzy soft binterior and few of their properties. We illustrate these concepts with examples and investigate some of their related properties. 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 with u v9u. We also study image and preimage of singlevalued neutrosophic soft topological kalgebras. By the same construction, every locally compact hausdorff space x is an open dense subspace of a compact hausdorff space having at most one point more than x.
We then present the foundations of the theory of soft topological spaces. Intuitionistic fuzzy neutrosophic soft topological spaces. Soft regular generalized bclosed sets in soft topological. Shabir and naz 2011 introduced and studied the notions of soft topological spaces, soft interior, soft closure and soft separation axioms. Soft generalized closed sets in soft topological spaces article pdf available in journal of theoretical and applied information technology 371. Pdf this paper demonstrates the redundancies concerning the increasing popular soft set approaches to general topologies. Pdf on soft topological spaces munazza naz academia. In this paper intuitionistic fuzzy neutrosophic soft topological spaces are introduced.
The objective of studing the current paper is to introduced a new class of soft open sets in soft topological spaces called soft open sets. Some properties and characterizations of soft semi local function are explored. Palaniappan and chandrasekhara rao introduced regular generalized closed sets in topological spaces. A decomposition of soft continuity in soft topological spaces. Second is to introduce soft product topology and study properties of soft. Zeba tarrannum and anil p narappanavar department of mathematic, global academy of technology, bengaluru 560098, karnataka, india. Soft topological space and topology on the cartesian product. There are many characterizations of the exponentiable spaces in the category top of topological spaces. Also we have shown that this soft topology is different from nazs soft topology and studied some basic properties of this new type of soft topology.
Then soft open sets are used to study some soft topological concepts. Pdf on binary soft topological spaces researchgate. Pdf the soft topological spaces and some their related concepts have been studied by shabir and naz in 1. Muhammad shabir and munazza naz introduced soft topological spaces and the notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms. In this paper we introduce fuzzy soft gclosed set, fuzzy soft. Subsequently, many works on fuzzy soft rings and fuzzy soft ideals 5, soft group 6, soft topological spaces 8, 10, fuzzy soft topological spaces 11, fuzzy soft open and closed sets 12, vector sum and scalar multiplication of soft sets, balanced and absorbing soft sets 14 have been done in this eld. 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.
Then we investigate some of the properties of fuzzy soft bopen sets. In this paper we give a new definition of soft topology using elementary union and elementary intersection although these operations are not distributive. Arya 23 introduced the notion of nearly paracompact spaces in topological spaces and studied their some 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. Here we use elementary complement of soft sets, though law of excluded middle is not valid. Dhana balan department of mathematics, alagappa govt. In this paper, we investigated the properties and characterizations of soft. Some basic concepts in soft generalized topological spaces are also defined and studied.
The soft closure of a soft set is defined which is, in fact, a. Department of mathematics, quaidiazam university, islamabad, pakistan. Noamanc adepartment of mathematics, faculty of science and arts najran university,k. Weak forms of soft open sets were first studied by chen 5. Soft generalized separation axioms in soft generalized. If f c is soft open in f b, then f c is soft open in f a. Pdf a note on soft topological spaces researchgate. Some basic properties of this new class of soft sets are investigated. Alsalem department of mathematics, college of science, basra, iraq.
This new class is defined over an initial universe and with a fixed set of parameters. The singleton p is called the support set of the soft point and. In this monograph we make the standing assumption that all vector spaces use either the real or the complex numbers as scalars, and we say real vector spaces and complex vector spaces to specify whether real or complex numbers are being used. Second is to introduce soft product topology and study properties of soft projection mappings. The aim of this article, we combine the fuzzy soft set and fuzzy topological space. In the present paper we introduce soft topological spaces which are defined over an initial universe with a fixed set of parameters. On binary soft topological spaces article pdf available in international journal of applied mathematics 306. Chapter 9 the topology of metric spaces uci mathematics. Pdf the purpose of this paper is to study soft sets and soft topological spaces.
A discrete fuzzy soft topological space is a fuzzy soft t 0 space since every fuzzy soft point e ha is a fuzzy soft open set in the discrete space. On soft almost paracompactness in soft topological space. Moreover, since f0is soft, the rst vertical map is surjective, hence the second one is surjective if and only if the third one is. Aug 19, 2011 the first aim of this study is to define soft topological spaces and to define soft continuity of soft mappings.
If xis a paracompact topological space and eis a soft sheaf of abelian groups on x, then hix. They got many important results in soft topological spaces. Pdf on exponentiable soft topological spaces sahand. Theoretical studies of soft topological spaces have also been researched by some authors in 2, 57, 15, 18, 20. Soft regular generalized bclosed sets in soft topological spaces. An embedding lemma in soft topological spaces 3 denoted byp.
Namely, we will discuss metric spaces, open sets, and closed sets. Further some of its properties and characterizations are established. We investigate behavior relative to union, intersection and soft subspaces of. An object x of a category c with finite limits is called exponentiable if the functor. The main purpose of this paper is to introduce soft compact soft generalized topological spaces as a generalization of compact spaces. 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. In this paper, we introduce some concepts such as soft connectedness, soft hausdorff spaces. Pdf soft semi local functions in soft ideal topological. Xis called a limit point of the set aprovided every open set ocontaining xalso contains at least one point a.
To contribute this research area, in this paper, we examine some properties on soft topological spaces such as neighborhood structure of a soft element and soft interior, soft closure, and soft cluster element and so on that are based on soft element definition that gives us a different perspective for development of soft set theory. Soft topological space is considered as mathematical tools for dealing with uncertainties, and a fuzzy topological space is a special case of the soft topological space. In this paper, we define a soft semi local function f, e. We introduce soft homogeneity as an extension of homogeneity in ordinary topological spaces. So, after the definition of fuzzy soft point, we are now in a position to introduce various separation axioms for fuzzy soft topological spaces and study their properties. We also introduce the concepts of soft interior point. Metricandtopologicalspaces university of cambridge. Pdf fuzzy soft bopen sets in fuzzy soft topological. The main goal is to point out that any soft topological space is homeomorphic to a topological space e u where. So the purpose of this note is, first, to point out some errors in remark 4 and example 9 of shabir and naz 2011, and second, to investigate properties of soft separation axioms. On idea of controlling soft gemset in soft topological space. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.
646 1123 1226 598 761 1414 725 423 1349 988 679 755 1436 186 1507 268 987 901 367 93 372 268 878 1551 1218 327 681 864 59 1354 539 924 238 613 304 479 911 1495 548