In 1970, levine9 introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. Balachandran, sundaram and maki 1 introduced and investigated the concept of generalized locally closed sets. For a subset a of x, the closure of a and the interior of a will. 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. The lower approximation of x with respect to r is the set of. On generalized topological spaces pdf free download. Decomposition of closed sets in supra topological spaces. Full text of on some locally closed sets and spaces in. Closed sets are fundamental objects in a topological space. Dontchev and maki have introduced the concept of generalized closed sets, in 1997 park et. In this paper weconsider a new class of generalized continuous. Informally, 3 and 4 say, respectively, that cis closed under. Introduction and preliminaries norman levine introduced and studied generalized closed briefly gclosed sets 11 and semiopen sets 12.
The collection of all locally closed sets of x, will be. Navaneethakrishnan and sivaraj 7 were introduced the concept of iglocallyclosed sets in ideal topological spaces. On generalized closed sets in generalized topological spaces. Saadati abstract we extend the notion of omega open set in ordinary topological spaces to generalized topological spaces. Regular sets in generalized topological spaces authors. The approach of the concept of semiregularization from a different perspective via the. Remind that each constructible set is a finite union of locally closed sets. In this paper we have introduced the concept of regular. Using generalized closed sets, dunham8 introduced the concept of the closure operator cl and a new topology and studied some of their properties. On generalized topological spaces artur piekosz abstract arxiv. R, pg student, nirmala college for women, coimbatore,tamil nadu. In the present work, a new class of sets called soft gclosed sets in soft topological spaces is introduced and some of their properties are studied. Some generalizations of locally closed sets iranian journal of.
Using generalized closed sets, dunham 1982 introduced the concept of generalized closure operator cl and obtained a class of topology, namely topology. Chinthamani vasthirani, 2assosiate professor, department of mathematics, sree sevugan annamalai college. In this paper another generalization of igclosed sets namely g i eclosed set is defined using semipre local function. According to bourbaki a subset of a topological space is locally closed if it is the intersection of an open set and a closed set.
Semi generalized bclosed sets in topological spaces 3. Generalized topological spaces and generalized open sets play a very impor. Regular weakly generalized locally closed sets in ideal topological spaces 1s. Maki 12 introduced the notion of sets in topological spaces. Vaidyanathaswamy 10 was given much importance to the topic and ideal topological space. Generalized closed sets in fuzzy topological spaces 35 proposition 3. Generalized topology of gtspace has the structure of frame and is closed under arbitrary unions and finite intersections modulo small subsets. This type of set was investigated by ekici and caldas 2004 under the name of jopen sets.
Finally we show that certain results of several publications on the concepts of weakness and strength of fuzzy generalized closed sets are. Also, we introduce a new separation axiom of the topological spaces, and we prove that every space is a space. Some weaker forms of decompositions in supra topological. Arts college, udumalpet, tamilnadu, india 2department of mathematics, psgr krishnammal college for women, coimbatore, tamilnadu, india abstract. In 1970, levine studied the concept of generalized closed sets and a class of topological spaces called t 12 space. Norman levine 7 introduced generalized closed briefly gclosed sets in 1970. Meena priyadarshini 1department of mathematics, kumaraguru college of technology, coimbatore, tamil nadu, india e mail. If the inline pdf is not rendering correctly, you can download the pdf file here. Decomposition of closed sets in supra topological spaces s. The aim of this paper is to introduce a new class of sets called delta generalized. J locally closed sets in bigeneralized topological spaces and study some of their properties. Steiner 5 gave a new type of generalized closed set in topological space called generalized b closed sets and study some of its fundamental properties.
Pdf generalized locally closed sets in ideal topological. Lo 12 jun 2009 in this paper a systematic study of the category gts of generalized topological spaces in the sense of h. Also it is shown that the class of mclosed sets is independent from the class of preclosed setsthe class of generalized closed setsthe class of g. Omega open sets in generalized topological spaces samer al ghour, wafa zareer department of mathematics and statistics, jordan university of science and technology, irbid 22110, jordan. Nano generalized locally closed sets and nglccontinuous functions in nano topological spaces 1. Sets and the associated closure operator, the special issue in.
The concepts of minimal open sets and maximal closed sets in topolog. Paper 1, section ii 12e metric and topological spaces. Abstract in this paper we introduce a new class of sets namely, gs closed sets, properties of this set are investigated and we. Navaneethakrishnan and others published generalized locally closed sets in ideal topological spaces find, read and cite all the research you need on researchgate. Andrijevic 1996 introduced a class of generalized open sets in a topological space called bopen sets. L, assistant professor, nirmala college for women, coimbatore, tamil nadu. A new class of generalized closed sets in topological spaces. Ais a family of sets in cindexed by some index set a,then a o c. Pdf locally gclosed sets in topological spaces researchgate. The aim of this work is to introduce operations on fuzzy topological spaces and to use them to study fuzzy generalized closed sets and fuzzy generalized open sets.
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. Also we discuss some of their properties and investigate the relations between the associated. This new class of sets widens the scope to do further research in the areas like bitopological spaces, smooth topological spaces and fuzzy soft topological spaces. Regular weakly generalized locally closed sets in ideal. Knebusch and their strictly continuous mappings begins. We introduce the notion of generalized topological space gtspace. Also, we introduce some characterizations and properties for these concepts. Nagaveni2 1,2 department of mathematics, coimbatore institute of technology, india abstract. Suppose h is a subset of x such that f h is closed where h denotes the closure of h.
A subset a of a topological space is said to be locally closed 4 if it is the intersection of an open set and a closed set. Generalized closed sets in generalized topological spaces 1m. Jeyaraman, 1 assistant professor, department of mathematics, rdm government arts college, sivagangai. A is a family of generalized topologies on a set x. Reily 6 introduced and studied different notions of generalized continuity and gave a decomposition. Roy and erdal ekici, on, closed sets in generalized topological spaces, method of functional analysis and topology, 172 2011, 174179. We investigate various classes of generalized closed fuzzy sets in topological spaces, namely, closed fuzzy sets and closed fuzzy sets. Minimal open sets on generalized topological space scielo. A new class of locally closed sets and locally closed.
Topological spaces, closed sets, separation axioms. Chapter 9 the topology of metric spaces uci mathematics. Levine 16 initiated the study of socalled generalized 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.
Alexandrov topology the topology of a space x is an alexandrov topology or is finitely generated if arbitrary intersections of open sets in x are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset. In this paper we study generalized closed sets in the sense of n. We define and study regular and locally closed sets in generalized topological spaces. A set is a set a which is equal to its kernel saturated set, i. Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy. Introduction to generalized topological spaces zvina. Rajarubi abstract in this paper, we introduce a new class of sets called. We also study the concept of generalizations of jlocally closed set in bgts. On generalized closed sets in generalized topological. Supra generalized locally closed sets in this section, we introduce the notions of supra generalized locally closed sets and discuss some of their properties.