Abstract. The aim of this paper is to study the class of θ-generalized closed sets, which is properly placed between the classes of generalized closed and θ-closed sets. Furthermore, generalized Λ-sets  are extended to θ-generalized Λ-sets and R 0 -, T 1/2 - and T 1 -spaces are characterized. The relations with other notions directly or indirectly connected with generalized closed sets are investigated. The notion of TGO-connectedness is introduced. Keywords and phrases. θ-generalized closed, θ-closure, Λ-set, TGO-connected.
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Regular open sets and strong regular open sets have been introduced and investigated by Stone  and Tong  respectively. Levine [8,9]. Biswas , Cameron, Sundaram and Sheik John , Bhattacharyya and Lahiri , Nagaveni , Pushpalatha , Gnanambal , gnanambal and balachandran , Palaniappan and Rao , Maki Devi and Balachandra , and Benchalli and Wali  ,Syed Ali Fathima introduced and investigated semi open sets, generalized closed sets, regular semi open sets, weakly closed sets, semi generalized closed sets, weakly generalized closed sets, strongly generalized closed sets, generalized pre-regular closed sets, regular generalized closed sets, α-generalized closed sets ,Rw-closed sets and #regular generalized closed sets respectively.
Abstract. The purpose of this note is to strengthen several results in the literature concerning the preservation of θ-generalized closed sets. Also conditions are established under which images and inverse images of arbitrary sets are θ-generalized closed. In this process several newweak forms of continuous functions and closed functions are developed.
Many researchers defined some basic notions on soft multi topological spaces and studied many properties. In this paper, we define soft multi generalized closed and open sets in soft multi topological spaces and studied their some properties. We introduce these concepts which are defined over an initial universe with a fixed set of parameters. We investigate behaviour relative to union, intersection and soft multi subspaces of soft multi generalized closed sets. Also, we investigate many basic properties of these concepts.
Atanassov  introduced the idea of intuitionistic fuzzy sets using the notion of fuzzy sets by Zadeh. Coker  introduced intuitionistic fuzzy topological spaces using the notion of intuitionistic fuzzy sets. Later this was followed by the introduction of intuitionistic fuzzy * generalized closed sets by Riya, V. M and Jayanthi, D  in 2017 which was simultaneously followed by the introduction of intuitionistic fuzzy * generalized continuous mappings  by the same authors. We have now extended our idea towards intuitionistic fuzzy * generalized closed mappings and discussed some of their properties.
bitopological space. Kelly  initiated the systematic study of such spaces. After the work of Kelly  various authors [2,3,7,8] turned their attention to generalization of various concepts of topology by considering bitopological spaces. The concept of generalized closed sets in bitopological spaces was introduced and investigated by T .
set (briefly g-closed) if Cl(A) ⊆ U whenever U ∈ τ and A ⊆ U. Generalized semiclosed  (resp., α-generalized closed , θ-generalized closed , generalized semi-preclosed , δ-generalized closed , ω-generalized closed [3, 11]) sets are defined by replacing the closure operator in Levine’s original definition by the semiclosure (resp., α-closure, θ-closure, semi-preclosure, δ-closure, ω-closure) operator.
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In 1970, Levine  introduced the concepts of generalized closed sets as a generalization of closed sets in topological spaces. Using generalized closed sets, Dunham  introduced the concept of the closure operator cl* and a new topology τ * and studied some of their properties. P. Bhattacharya and B.K. Lahiri introduced the concept of semi generalized closed sets in topological spaces. J. Dontchev  introduced generalized semi-open sets, H. Maki, R. Devi and K. Balachandran  introduced generalized α -closed sets in topological spaces.
 Arya,S.P., and Nour,T., Characterizations of S-normal spaces, Indian J.pure Appl. Math., Vol.21, pp.717-719,1990.  Bhattacharyya,P., and Lahiri, B.K., Semi-generalized closed sets in topology, Indian J. Math., Vol.29, pp. 376-382,1987.  Dontchev,J.,On generalizing semi –pre open sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., Vol.16,pp. 35-48,1995.  Levine, N.,Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, Vol.70,pp.36-41,1963.  Levine, N.,Generalized closed sets in topological spaces, Rend. Circ. Mat. Palermo, Vol.19(2),pp.89-96,1970.
N. Levine  introduced generalized closed sets in general topology as a generalization of closed sets. This concept was found to be useful and many results in general topology were improved. Many researchers like Balachandran, Sundaram and Maki , Bhattacharyya and Lahiri , Arockiarani , Dunham , Gnanambal , Malghan , Palaniappan and Rao , Park  Arya and Gupta  and Devi , Benchalli and wali  have worked on generalized closed sets, their generalizations and related concepts in general topology. In this paper, we define and study the properties of (1,2)*-regular generalized α-closed sets (briefly (1,2)*-rgα-closed)in bitopological spaces.
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Levine introduced generalized closed sets in topological space. Veera kumar introduced and studied # generalized semi closed set, Naga veni.N introduced weakly generalized closed sets in topological spaces. Rajesh and Lellis Thivagar g - closed set, Saied Jafari,Lellis Thivagar and Nirmala Rebecca Paul introduced and studied g -closed set. The authors introduced and studied the weaker form of g -WG closed set. Dontchev  introduced and investigated a new notion of continuity is called contra-continuity. Jafari and Noiri ,, introduced new generalization of contra continuity called contra super continuity, contra-α-continuity and contra-pre continuity. The purpose of the present paper is to introduce and investigate some of the properties of contra g ~ wg – continuous functions, contra g ~ wg-irresolute functions and we obtain characterization of contra g ~ wg-continuous function.
Levine  introduction and investigated the concept of generalized closed sets in topological space .Arya and Nour defined generalized semi open [briefly gs- open] sets using semi open sets. In 1987 Bhattacharya and Lahiri  introduced the class of semi – generalized closed sets (sg- closed sets) Balachandran  introduced generalized continuous maps in topological spaces. Homomorphism plays a very important role in topology.
The study of semi-open (briefly s-open) sets in a topological spaces was initi- ated by N.Levine in 1963 and also he generalized the concept of closed sets to generalized closed(briefly g-closed) sets in 1970. Bhattacharya and Lahiri generalized the concept of closed sets to semi-generalized closed(briefly sg- closed) sets in 1987. O.Njastad introduced αsets(called as α-closed sets).
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Adel.M.AL.Odhari derived the concept of infra topological space. In 1970, Levine  introduced the concept of generalized closed sets which formed a strong tool in the characterization of topological spaces. In 1996, D.Andrijevic‟ introduced and studied the class of generalized open sets in a topological space called b-open sets .This class of sets contained in the class of -open sets , and contains all semi-open sets , and all pre-open sets. During recent years‟ different types of generalized closed sets were introduced and studied by many topologist [8,9,10,12,13,15,16,17]. Nagaveni introduced weakly generalized closed sets and investigated the properties of these sets. Bhattacharyya & Lahiri  introduced semi generalized closed set and Arya &Nour  (1990) introduced generalized semi closed set in Topological space.
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3. FORMS OF NANO REGULAR GENERALIZED CLOSED SETS AND NANO GENRALIZED REGULAR CLOSED SETS In this section, we introduce Nano regular generalized closed set and Nano generalized regular closed set and investigate some of their properties .
concept of generalized open sets and b-open sets respectively in topological spaces. The class of b-open sets is contained in the class of semi pre open sets and contains the class of semi open and the class of preopen sets. Since several researches were done and the notion of generalized semi closed, generalized preclosed and generalized semi-pre open sets were investigated. In 1968 Zaitsev defined π-closed sets. Later Dontchev and Noiri introduced the notion of πg-closed sets. The idea of πgb-closed sets was introduced by D. Sreeja and C. Janaki . Dhanya. R, A. Parvathi, Introduced “ On πgb ⃰-closed sets in Topological spaces”,. This paper is an attempt to highlight a new type of generalized closed sets called (1,2) ⃰-π generalized b ⃰ -closed, called (1,2) ⃰ - πgb ⃰ -continuous maps and (1,2) ⃰ - πgb ⃰ - irresolute maps in bitopological spaces. These findings results in procuring several characterizations of this class. It analyses its bitopological properties. The application of this set leads to the introduction of a new space called (1,2) ⃰ - πgb ⃰ - space.
The idea of grill on a topological space was first introduced by Choquet  in the year 1974. The grill concept depends upon two operators namely Ф and Ψ. We observe that the grill concept is the most powerful tool for supporting many things like nets and filters. A number of theories and feature helped to expand the topological structure. This expansion of topological space is used to measure the technique of describing other than quantity, such as love, beauty and intelligence,etc... Grill concept is also used to expand the topological structure. This would change in lower approximation, upper approximation and boundary region which give new goal in Nano topological spaces. The nano topology concept was given by Lellis Thivagar and Carmel Richard . Many authors [2,10] introduced the concept of Nano generalized alpha closed sets, Nano alpha generlized closed sets, Nano semi generalized closed sets, Nano generalized alpha interior and Nano generalized alpha closure in Nano topological spaces. Azzam A.A  introduced a new concept Grill nano generalized closed sets in grill nano topological spaces. Recently in  Chitra.V and Jayalakshmi.S introduced G -NSg closed sets and G- NgS closed sets and studied some properties in Grill nano topological spaces. Now our aim in this paper is to introduce the concept grill in nano generalized alpha closed sets and nano alpha generalized closed sets and their respective interiors and closures are also introduced and discuss some of their basic properties in Grill nano topological spaces.
Levine introduced generalized closed sets in topology as a generalization of closed sets. This concept was found to be useful and many results in general topology were improved. Many researchers like Arya et al, Balachandran et al, Bhattarcharya et al, Arockiarani et al, Gnanambal , Nagaveni and Palaniappan et al have worked on generalized closed sets. Andrjivic gave a new type of generalized closed set in topological space called b closed sets. A.A.Omari and M.S.M. Noorani  made an analytical study and gave the concepts of generalized b closed sets in topological spaces.
In 1983, Mashhour et al.  introduced the supra topological spaces and studied S-continuous functions and S*- continuous functions. In 2011, Ravi et al.  introduced and investigated several properties of supra generalized closed sets, supra sg-closed sets and gs-closed sets in supra topological spaces. In topological space the arbitrary union condition is enough to have a supra topological space. Here every topological space is a supra topological space but the converse is not always true. Many researchers are Introducing many new notions and investigating the properties and characterizations of such new notions In this paper we introduced Supra*g- closed set in supra topological space and properties and characterization are discussed in details
The notion of closed set is fundamental in the study of topological spaces. In 1970, Levine  introduced the concept of generalized closed sets in topological spaces by comparing the closure of a subset with its open supersets. Further the study of g-closed sets was continued by Dunham and Levine . Maximal open sets and Minimal open sets were studied and introduced by Nakaoka and Oda [3,4,5]. Benchalli, Banasode and Siddapur introduced and studied generalized minimal closed sets in topological spaces . Further Banasode and Desurkar introduced and studied generalized minimal closed sets in bitopological spaces .