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 [16] 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 [18] and Tong [20] respectively. Levine [8,9]. Biswas[3] , Cameron[4], Sundaram and Sheik John [17], Bhattacharyya and Lahiri [2], Nagaveni [12], Pushpalatha [16], Gnanambal [6], gnanambal and balachandran [7], Palaniappan and Rao [13], Maki Devi and Balachandra [10], and Benchalli and Wali [1] ,Syed Ali Fathima[19] 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 [1] introduced the idea of intuitionistic fuzzy sets using the notion of fuzzy sets by Zadeh. Coker [2] 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 [7] in 2017 which was simultaneously followed by the introduction of intuitionistic fuzzy * **generalized** continuous mappings [8] 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 [5] initiated the systematic study of such spaces. After the work of Kelly [5] 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 [7].

set (briefly g-**closed**) if Cl(A) ⊆ U whenever U ∈ τ and A ⊆ U. **Generalized** semiclosed [6] (resp., α-**generalized** **closed** [7], θ-**generalized** **closed** [8], **generalized** semi-preclosed [9], δ-**generalized** **closed** [10], 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 [6] introduced the concepts of **generalized** **closed** sets as a generalization of **closed** sets in topological spaces. Using **generalized** **closed** sets, Dunham [5] 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[3] introduced the concept of semi **generalized** **closed** sets in topological spaces. J. Dontchev [4] introduced **generalized** semi-open sets, H. Maki, R. Devi and K. Balachandran [9] introduced **generalized** α -**closed** sets in topological spaces.

[2] Arya,S.P., and Nour,T., Characterizations of S-normal spaces, Indian J.pure Appl. Math., Vol.21, pp.717-719,1990. [3] Bhattacharyya,P., and Lahiri, B.K., Semi-**generalized** **closed** sets in topology, Indian J. Math., Vol.29, pp. 376-382,1987. [4] Dontchev,J.,On generalizing semi –pre open sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., Vol.16,pp. 35-48,1995. [5] Levine, N.,Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, Vol.70,pp.36-41,1963. [6] Levine, N.,**Generalized** **closed** sets in topological spaces, Rend. Circ. Mat. Palermo, Vol.19(2),pp.89-96,1970.

N. Levine [1] 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 [3], Bhattacharyya and Lahiri [4], Arockiarani [2], Dunham [5], Gnanambal [6], Malghan [7], Palaniappan and Rao [8], Park [9] Arya and Gupta [12] and Devi [13], Benchalli and wali [11] 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[9] introduced **generalized** **closed** sets in topological space. Veera kumar[21] introduced and studied # **generalized** semi **closed** set, Naga veni.N[15] introduced weakly **generalized** **closed** sets in topological spaces. Rajesh and Lellis Thivagar[17] g - **closed** set, Saied Jafari,Lellis Thivagar and Nirmala Rebecca Paul[18] introduced and studied g -**closed** set. The authors[11] introduced and studied the weaker form of g -WG **closed** set. Dontchev [4] introduced and investigated a new notion of continuity is called contra-continuity. Jafari and Noiri [6],[7],[8] 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 [7] introduction and investigated the concept of **generalized** **closed** sets in topological space .Arya and Nour[1] defined **generalized** semi open [briefly gs- open] sets using semi open sets. In 1987 Bhattacharya and Lahiri [3] introduced the class of semi – **generalized** **closed** sets (sg- **closed** sets) Balachandran [2] 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[7] in 1963 and also he **generalized** the concept of **closed** sets to **generalized** **closed**(briefly g-**closed**) sets[8] in 1970. Bhattacharya and Lahiri[3] **generalized** the concept of **closed** sets to semi-**generalized** **closed**(briefly sg- **closed**) sets in 1987. O.Njastad[14] introduced αsets(called as α-**closed** sets).

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Adel.M.AL.Odhari[1] derived the concept of infra topological space. In 1970, Levine [12] introduced the concept of **generalized** **closed** sets which formed a strong tool in the characterization of topological spaces. In 1996, D.Andrijevic‟[2] 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 [3], and contains all semi-open sets [13], and all pre-open sets[20]. 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[15] introduced weakly **generalized** **closed** sets and investigated the properties of these sets. Bhattacharyya & Lahiri [7] introduced semi **generalized** **closed** set and Arya &Nour [6] (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[19] defined π-**closed** sets. Later Dontchev and Noiri[9] introduced the notion of πg-**closed** sets. The idea of πgb-**closed** sets was introduced by D. Sreeja and C. Janaki [17]. Dhanya. R, A. Parvathi, Introduced “ On πgb ⃰-**closed** sets in Topological spaces”,[8]. 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 [4] 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 [11]. 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 [1] introduced a new concept Grill nano **generalized** **closed** sets in grill nano topological spaces. Recently in [3] 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[5], Balachandran et al[6], Bhattarcharya et al[7], Arockiarani et al[4], Gnanambal [8], Nagaveni[14] and Palaniappan et al[15] have worked on **generalized** **closed** sets. Andrjivic[3] gave a new type of **generalized** **closed** set in topological space called b **closed** sets. A.A.Omari and M.S.M. Noorani [2] made an analytical study and gave the concepts of **generalized** b **closed** sets in topological spaces.

In 1983, Mashhour et al. [1] introduced the supra topological spaces and studied S-continuous functions and S*- continuous functions. In 2011, Ravi et al. [3] 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 [1] 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 [1]. 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 [2]. Further Banasode and Desurkar introduced and studied **generalized** minimal **closed** sets in bitopological spaces [7].