Definition and examples, topological spaces duration. A basis b for a topological space x is a set of open sets, called basic open sets, with the following properties. In this research paper, a new class of open sets called ggopen sets in topological space are introduced and studied. Because of this theorem one could define a topology on a space using closed sets instead of open sets. Oct 20, 2018 open sets and closed sets in a topological space, topology, lecture1 arvind singh yadav,sr institute for mathematics. Also some of their properties have been investigated. A generalized semipreclosed gspclosed set if spcla. Bc open subsets of a topological space is denoted by. 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. The concept of generalized closed sets and generalized open sets was first. Abstract in this paper we introduce a new class of sets namely, gsclosed sets, properties of this set are investigated and we. Also we discuss some of their properties and investigate the relations between the associated. Ais a family of sets in cindexed by some index set a,then a o c. Maki 12 introduced the notion of sets in topological spaces.
The complements of the above open sets are called their respective closed sets. In this paper, we have introduced a new class of sets called bg closed sets in topological spaces. 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. Using generalized closed sets, dunham 1982 introduced the concept of generalized closure operator cl and obtained a class of topology, namely topology. Informally, 3 and 4 say, respectively, that cis closed under. A subset a of x is said to be bg closed if bcla u whenever a u and u is g open in x. We also present and study new separation axioms by using the notions of.
Finite spaces have canonical minimal bases, which we describe next. Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy. 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. The concepts of z open set and z continuity introduced by mubarki. Introduction in 1970, levine9 introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. Later in 1996 andrjivic gave a new type of generalized closed set in topological space called b closed sets. In general topological spaces a sequence may converge to many points at the same time. Generalized alpha closed sets in neutrosophic topological spaces. Closed sets 34 open neighborhood uof ythere exists n0 such that x n. L, assistant professor, nirmala college for women, coimbatore, tamil nadu. Also, we introduce a new separation axiom of the topological spaces, and we prove that every space is a space. Soft regular generalized bclosed sets in soft topological spaces. This paper committed to the investigation of neutrosophic topological spaces. The investigation on generalization of closed set has lead to signi.
In this research paper, a new class of open sets called gg open sets in topological space are introduced and studied. Closed sets in topological spaces article pdf available in international journal of mathematical analysis 839. Also we introduce new functions semi open and semi closed functions. 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 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. Department of mathematics, sri eshwar college of engineering, coimbatore641 202, tamil nadu, india, a. Closed sets in topological spaces semantic scholar. Recently we introduced semi open sets and semi continuity to obtain decomposition of continuity. 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. Generalized closed sets via neutrosophic topological spaces.
Andrijevic 1996 introduced a class of generalized open sets in a topological space called b open sets. We investigate various classes of generalized closed fuzzy sets in topological spaces, namely, closed fuzzy sets and closed fuzzy sets. The open sets in a topological space are those sets a for which a0. International journal of computer applications 0975 8887 volume 125 no. On generalized closed sets and generalized preclosed sets in. A set is a set a which is equal to its kernel saturated set, i. In w 2 we shall define generalized closed written henceforth as gclosed sets and characterize them as. On generalized closed sets and generalized preclosed sets. In this paper, we obtain several characterizations of semi open sets and semi continuous functions. In this paper generalized alpha closed sets and generalized alpha open sets are presented. 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. The closure of a is the union of the interior and boundary of a, i.
We also introduce ggclosure, gginterior, ggneighbourhood, gglimit points. Paper 1, section ii 12e metric and topological spaces. 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. Pdf closed sets in topological spaces iaset us academia. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. Definition for a topological space x, the topology is defined by g. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals. New class of generalized closed sets in supra topological spaces. Preregular spopen sets in topological spaces scielo. Topology is a classical subject, as a generalization topological spaces many type of topological spaces introduced over the year.
Pdf bcopen sets in topological spaces researchgate. Chapter 3 semi generalized bclosed sets in topological spaces. N 7 introduced strong continuity in topological spaces. Generalized closed sets via neutrosophic topological spaces a. Suppose h is a subset of x such that f h is closed where h denotes the closure of h. Generalized closed sets in topological spaces in this section, we introduce the concept of. Soft semi open sets and its properties were introduced and studied by bin chen4. Pdf closed sets in topological spaces researchgate. One often says \x is a topological space so mean that there is t such that x. 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. The elements of g are called neutrosophic open sets. Paper open access neutrosophic generalized bclosed sets in. The study of generalized closed sets in a topological space was initiated by levine in 7. Semi generalized b closed sets in topological spaces 3.
This type of set was investigated by ekici and caldas 2004 under the name of j open sets. R, pg student, nirmala college for women, coimbatore,tamil nadu. Neutrosophic set, generalized neutrosophic set, neutrosophic topology introduction and preliminaries. The function f is called open if the image of every open set in x is open in y. Given topological spaces x and y, a function f from x to y is continuous if the preimage of every open set in y is open in x. Dontchev and maki have introduced the concept of generalized closed sets, in 1997 park et. Chapter 9 the topology of metric spaces uci mathematics. The open and closed sets of a topological space examples 1.