The elements of g are called neutrosophic open sets. The investigation on generalization of closed set has lead to signi. In w 2 we shall define generalized closed written henceforth as gclosed sets and characterize them as. In this paper, we obtain several characterizations of semi open sets and semi continuous functions. Also some of their properties have been investigated. 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. Ais a family of sets in cindexed by some index set a,then a o c. The main purpose of this paper is to introduce and study new classes of soft closed sets like soft rgb closed, soft rg closed, soft gpr closed, soft gb closed, soft gsp closed, soft g closed, soft g b closed, and soft sgb closed sets in soft topological spaces. This applies, for example, to the definitions of interior, closure, and frontier in pseudometric spaces, so these definitions can also be carried over verbatim to a topological space. Preregular spopen sets in topological spaces scielo. A basis b for a topological space x is a set of open sets, called basic open sets, with the following properties. We investigate various classes of generalized closed fuzzy sets in topological spaces, namely, closed fuzzy sets and closed fuzzy sets. Later in 1996 andrjivic gave a new type of generalized closed set in topological space called b closed sets. Using generalized closed sets, dunham 1982 introduced the concept of generalized closure operator cl and obtained a class of topology, namely topology.
Topology is a classical subject, as a generalization topological spaces many type of topological spaces introduced over the year. In this paper generalized alpha closed sets and generalized alpha open sets are presented. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. In this research paper, a new class of open sets called ggopen sets in topological space are introduced and studied. We also present and study new separation axioms by using the notions of. Also we discuss some of their properties and investigate the relations between the associated. Soft regular generalized bclosed sets in soft topological spaces. Bc open subsets of a topological space is denoted by. Closed sets in topological spaces article pdf available in international journal of mathematical analysis 839.
Andrijevic 1996 introduced a class of generalized open sets in a topological space called b open sets. N 7 introduced strong continuity in topological spaces. Definition for a topological space x, the topology is defined by g. The concept of generalized closed sets and generalized open sets was first. A generalized semipreclosed gspclosed set if spcla. This paper committed to the investigation of neutrosophic topological spaces. The closure of a is the union of the interior and boundary of a, i. Also we introduce new functions semi open and semi closed functions. We also introduce ggclosure, gginterior, ggneighbourhood, gglimit points. Pdf closed sets in topological spaces researchgate.
In this research paper, a new class of open sets called gg open sets in topological space are introduced and studied. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals. Generalized closed sets in topological spaces in this section, we introduce the concept of. Introduction in 1970, levine9 introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. A subset a of x is said to be bg closed if bcla u whenever a u and u is g open in x. New class of generalized closed sets in supra topological spaces. Oct 20, 2018 open sets and closed sets in a topological space, topology, lecture1 arvind singh yadav,sr institute for mathematics. The open sets in a topological space are those sets a for which a0.
The complements of the above open sets are called their respective closed sets. Preliminaries definition for the subset a of a topological space x the generalized closure operator cl is defined by the intersection of all gclosed sets containing a. Soft semi open sets and its properties were introduced and studied by bin chen4. In this case, the pair z,gis called a neutrosophic topological space nts for short and any neutrosophic set in g is known as neutrosophic open set nos 2z. In general topological spaces a sequence may converge to many points at the same time. Informally, 3 and 4 say, respectively, that cis closed under. One often says \x is a topological space so mean that there is t such that x. On generalized closed sets and generalized preclosed sets in. Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy.
Definition and examples, topological spaces duration. Semi generalized b closed sets in topological spaces 3. The study of generalized closed sets in a topological space was initiated by levine in 7. Maki 12 introduced the notion of sets in topological spaces. Department of mathematics, sri eshwar college of engineering, coimbatore641 202, tamil nadu, india, a. L, assistant professor, nirmala college for women, coimbatore, tamil nadu. Generalized alpha closed sets in neutrosophic topological spaces. On generalized closed sets and generalized preclosed sets. Paper 1, section ii 12e metric and topological spaces. On generalized closed sets and generalized preclosed sets in neutrosophic topological spaces wadei alomeri 1,, and saeid jafari 2, 1 department of mathematics, albalqa applied university, salt 19117, jordan 2 department of mathematics, college of vestsjaelland south, herrestraede 11, 4200 slagelse, denmark. In this paper, we have introduced a new class of sets called bg closed sets in topological spaces. Chapter 3 semi generalized bclosed sets in topological spaces.
Finite spaces have canonical minimal bases, which we describe next. The concepts of z open set and z continuity introduced by mubarki. Dontchev and maki have introduced the concept of generalized closed sets, in 1997 park et. Recently we introduced semi open sets and semi continuity to obtain decomposition of continuity. Closed sets in topological spaces semantic scholar. Generalized closed sets via neutrosophic topological spaces. Pdf closed sets in topological spaces iaset us academia. Paper open access neutrosophic generalized bclosed sets in. Pdf bcopen sets in topological spaces researchgate.
The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces. Closed sets 34 open neighborhood uof ythere exists n0 such that x n. Chapter 9 the topology of metric spaces uci mathematics. 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. Neutrosophic set, generalized neutrosophic set, neutrosophic topology introduction and preliminaries. Nandhini 2 abstract in this paper, we have introduced the notion of generalized closed sets in neutrosophic topological spaces and studied some of their basic properties. Whenever a 2 rn and r is a positive real number we let uar fx 2 rn. This type of set was investigated by ekici and caldas 2004 under the name of j open sets. A set is a set a which is equal to its kernel saturated set, i. Suppose h is a subset of x such that f h is closed where h denotes the closure of h. Also, we introduce a new separation axiom of the topological spaces, and we prove that every space is a space. Because of this theorem one could define a topology on a space using closed sets instead of open sets.
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