On a generalized closed sets in ideal topological spaces. A subset a of x is said to be ideal generalized closed set briefly ig closed set if a u whenever a. Alternative characterizations of topological spaces closed. Khan and noiri 3 have introduced semilocal functions in ideal topological spaces. The intersection of any collection of closed sets is also closed. Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy. In this paper, we introduce and study the concept of generalized closed sets with respect to an ideal in an ideal bitopological space. The concept of generalized closed sets and generalized open sets was first. A subset a of a space x is called a delta generalized. Generalized closed sets in topological spaces in this section, we introduce the concept of. 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.
Informally, 3 and 4 say, respectively, that cis closed under. The investigation on generalization of closed set has lead to significant contribution to the theory of separation axiom, generalization of continuity and irresolute functions. Certain new classes of generalized closed sets and their. Firstly the notion of igclosed set is given by dontchev et al. Some of their properties and characterizations are studied. The closed subsets of a topological space satsify the following properties. Further, we introduce the notions of interior, closure, boundary, exterior and study some of their properties. Chapter 9 the topology of metric spaces uci mathematics. A collection i px is called an ideal on x if it satis. Introduction the concept of generalized closed sets as a generalization of closed sets in topological spaces was introduced by levine4 in 1970. The concept of generalized closed sets in bitopological spaces was introduced by sundaram.
A sufficient condition for a soft gclosed set to be a soft closed is also presented. The class of such sets is found to lie strictly between the class of all closed sets and that of generalized closed sets of levine 5. International journal of mathematics trends and technology. Closed sets in neutrosophic topological spaces university of new mexico n. One often says x is a topological space so mean that there is t such that x,t is a topological space. The notion of ideals in general topological spaces is treated in the classic text by kuratowski 4 and also in 10. On regular generalized open sets in topological space citeseerx. We also introduce the notion of generalized preregular continuity and study its. We will consider the question of when some classes of generalized closed sets coincide.
In this paper, the concept of fuzzy generalized regular star closed sets is introduced and studied. In 1970, levine9 introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. Closed sets, hausdor spaces, and closure of a set 9 8. The open sets in a topological space are those sets a for which a0. For a subset a of x, the closure of a and the interior of a will. On generalized closed sets in generalized topological spaces. Suppose h is a subset of x such that f h is closed where h denotes the closure of h. Using generalized closed sets, dunham 1982 introduced the concept of. Using these axioms, another way to define a topological space is as a set x together with a collection. A subset a of a generalized topological space we recall some notion defined in 1, 2. A set is a set a which is equal to its kernel saturated set, i. Pdf closed sets in topological spaces researchgate. Irjet nano generalized delta semi closed sets in nano.
Evidently specifying the open subsets is equivalent to specfying the closed subsets. We also introduce ggclosure, gginterior, ggneighbourhood, gglimit points. The aim of this paper is to introduce a new class of sets namely rg closed sets in topological spaces and study some basic properties. This type of set was investigated by ekici and caldas 2004 under the name of jopen sets. We will provide characterizations of extremally disconnected spaces and sgsubmaximal spaces by using various kinds of generalized closed sets. Pdf soft generalized closed sets in soft topological spaces. The purpose of this paper introduce and study the notions of. Decompositions of regular open sets and regular closed sets are provided using regular sets. Because of this theorem one could define a topology on a space using closed sets instead of open sets. An ideal on a set x is a nonempty collection of subsets of x with heredity property which is also closed under finite unions. The concept of generalized closed sets as a generalization of closed sets in topological spaces was introduced by levine4 in 1970. Also, we introduce a new separation axiom of the topological spaces, and we prove that every space is a space.
Also we discuss some of their properties and investigate the relations between the associated. Maki 12 introduced the notion of sets in topological spaces. Many variations of closed sets were introduced and investigated. Indeed ideals are very important tools in general topology.
This concept was found to be useful and many results in general topology were improved. Suguna selvarani department of mathematics, svs college of engineering, coimbatore, tamilnadu abstract. Introduction in 1990, jankovic and hamlett 3, have initiated the application ideal topological spaces. The union of any finite number of closed sets is also closed.
Noiri gave a new type of generalized closed sets in topological space called gp. Chapter 3 semi generalized bclosed sets in topological spaces. In this paper, it will be shown that in ideal topological spaces, a set which is both closed set and semipre dense set is a. Some properties are proved and their relations with different fuzzy sets in fuzzy topological spaces are investigated. Open sets, closed sets, bases, and subbases chapter. 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. Properties of these sets are investigated for topological spaces and generalized topological spaces. Pdf on generalized closed sets in generalized topological spaces. Sheik john 24 veera kumar 26 introduced the notion of closed sets g closed sets. On generalized closed sets in topological spaces dr. Also, some lower separation axioms weaker than t 1 are investigated.
Parimelazhagan2 1research scholar,karpagam university,india 2department of science and humanities, karpagam college of engineering, anna university, india abstract. Using generalized closed sets, dunham 1982 introduced the concept of generalized closure operator cl and obtained a class of topology, namely topology. The authors introduce new type of closed set named gp closed set in the topological spaces. Keywords closed sets, closed sets, continuous, irresolute, continuous maps, irre solute maps i. Rodgio and et al2 introduce d closed sets in topological spaces and studied their properties. The collection of all interior points of set e is the interior of e, denoted inte. We give some applications of 1 closed set and 1open set in. In 1970, levine studied the concept of generalized closed sets and a class of topological spaces called t. A new class of generalized closed sets in topological spaces. X is an interior point of set e provided there is a neighborhood of x that is contained in e.
Irgclosed, sawirgclosed, wrgiclosed, ideal topological spaces 1. Later on jin han park3 studied the concept of regular generalized closed in fuzzy topological space. The elements of in this paper the idea of locally closed sets in generalized topological space is. Semiconnectedness is characterized by using regular sets. In this paper, the authors introduce and study the concept of a new class of closed sets called weakly generalized e.
On generalized closed sets in generalized topological. Noiri gave a new type of generalized closed sets in topological space called gp closed sets. Saffiotti, behavioural navigation on topologybased maps, in proceedings of the 8th symposium on robotics with applications, maui, hawaii, usa, 2000. Kopperman, some topologybased image processing algorithms, annals of the new york academy of sciences, vol. Using generalized closed sets, dunham8 introduced the concept of the closure operator cl and a new topology and studied some of their properties. Between closed sets and z closed sets in topological spaces c. In w 2 we shall define generalized closed written henceforth as gclosed sets and characterize them as. Semi generalized bclosed sets in topological spaces 3. In this paper, it will be shown that in ideal topological spaces, a set which is both closed set and semipre dense set is a closed set. Introduction in the year 1963, levine of semi open sets and semi continuity in topological spaces tried to generalize topology by replacing open sets with semi open sets. The concepts of zopen set and zcontinuity introduced by mubarki. Pdf in this paper, we introduce several types of generalized closed sets in generalized topological spaces gtss.
Generalized closed sets with respect to an ideal european. The aim of this paper is to investigate the properties of these sets in the ideal topological spaces. A new class of locally closed sets and locally closed. Meena priyadarshini 1department of mathematics, kumaraguru college of technology, coimbatore, tamil nadu, india e mail. Between closed sets and closed sets in topological spaces. Semi generalized b closed sets in topological spaces 3. Soft generalized closed sets in soft topological spaces article pdf available in journal of theoretical and applied information technology 371. Generalized closed sets with respect to an ideal in.
Veerakumar 15 defined gclosed sets in topological spaces and studied their properties. Sheik john 24 veera kumar 26 introduced the notion of closed sets gclosed sets. Using generalized closed sets, dunham 1982 introduced the concept of generalized closure operator cl and. In this research paper, a new class of open sets called ggopen sets in topological space are introduced and studied. In this paper the structure of these sets and classes of sets are investigated, and some applications are given. Ti i 0, 1 2,1,2 spaces are characterized them using. Mar 11, 2009 an ideal on a set x is a nonempty collection of subsets of x with heredity property which is also closed under finite unions. Paper 1, section ii 12e metric and topological spaces.
By definition of a subset a of a topological space x is called generalized closed briefly, gclosed set if cla. Nagaveni assistant pro fessor coimbatore institute of technology, coimbatore, india. Also some of their properties have been investigated. Generalized closed sets in ideal topological spaces. 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. The aim of this paper is to introduce and study a new class of generalized closed sets called g p closed sets in topological spaces using gpclosed sets. Nano generalized delta semi closed sets in nano topological. Ig closed sets, semi i closed set, pre i closed set. Andrijevic 1996 introduced a class of generalized open sets in a topological space called bopen sets.
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