Arenas et al. [1] introduced the notion of -closed sets as a generalization of locally closed sets. In this paper, we introduce the notions of -locally closed sets,  -closed sets and  g-closed sets and obtain some decompositions of closed sets and continuity in topological spaces.

In this paper especial subsets of an R-module M called "pseudo multiplicatively closed" (p.m.c) are defined and many of the results such as the important theorem of Cohen for rings are proved for modules. Saturated p.m.c for modules is also defined and it shows that it is a generalization of the saturated p.m.c for rings. If S is a p.m.c subset of an R-module M, then we show that the smallest saturated p.m.c exists and is [M]- Up pns=,p PЄSpec(M). Modules of quotients are also defined and realized a relationship between p.m.c subsets of M and p.m.c subsets of Modules of quotients. Also if N is a submodule, then there is one to one correspondence between p.m.c subsets of M not containing 0 and p.m.c subsets of M disjoint from N.

Background and Aim: Different surgical techniques have been proposed for crown exposure among patients under orthodontic treatment. The aim of this study was to investigate periodontal indices in closed flap operation technique used for impacted teeth crown exposure.Materials and Methods: In this descriptive study (2001-2003) 20 patients (15 females and 5 males), age ranging from 15 to 20 years, were participated. Following the performance of closed flap operation technique, in 12 patients impacted canine and central incisors were exposed and treated orthodontically. Periodontal parameters including sulcular depth, width of keratinized gingiva and attachment loss were recorded. Results: The mean of gingival sulcus depth was 1.50±0.43, the means of keratinized gingival in central incisors and canines were 3.07±0.53 and 2.30±0.42, respectively. In one patient, attachment loss was reported.Conclusion: Considering the optimal sulcular depth and keratinized gingiva, the closed flap operation technique seems suitable.

Introduction: The Preferential method in Fixation of some long bone fractures consists of using intramedullary nailing. The benefit of this method lies in its better biomechanical stability، as well as preventing additional surgical trauma to the fracture site. Despite these benefits، few surgeons conduct this method as it is time-consuming، risky (Performing x-ray fluoroscopy) and also tedious. Method: Seven male patients with diaphyseal femoral fractures، with an average age of 32. 28، underwent closed femoral nailing surgery making use of the innovative set. The duration of operation، fracture type and fracture site were evaluated. The innovative solution: To align the bone fragments and make the guidance stick cross the fracture site closely، the present close reduction set was designed to make crossing the guidance stick through the fracture site possible on the normal bed of operation room in a short period of time without fluoroscopy، with the least possible extra trauma and the least degree of annoyance in the reduction stage. Results: All the patients with diaphyseal femoral fracture، without any peripheral fractures، underwent nailing surgery، while total time of the surgery was 116. 57 minutes on average; and pure time of the innovative set application and passing guidance stick to the bottom of the distal canal was 21. 85 minutes. Conclusion: The results indicated that closed intramedullary nailing of femoral fractures without performing Fluoroscope، using the aforementioned innovative set is possible for the patients with utter convenience in an appropriate time period.

The main purpose of this paper is to introduce and study new classes of soft closed sets like soft regular generalized b-closed sets in soft topological spaces (briefly soft-rgb-closed set) Moreover, soft rga-closed, soft gpr-closed, soft gb-closed, soft gsp-closed, soft ga-closed, soft gab-closed, and soft sgb-closed sets in soft topological spaces are introduced in this paper and we investigate the relations between soft rgb-closed set and the associated soft sets. Also, the concept of soft semi-regularization of soft topology is introduced and studied their some properties. We introduce these concepts which are defined over an initial universe with a fixed set of parameters.

Considering the e-kernel defined by ¨, Ozko¸, c and Ayhan [18] in a topological space, a new type of generalized closed set is studied through this article. The aim of this paper is to introduce a new class of sets called geɅ,-closed sets and geɅ,-open sets in a topological space and to study their properties and characterizations.

. The class of strongly Ɵcl - continuous functions is considered in this paper. Studying about basic properties of strongly Ɵcl - continuous functions, it is observed that these properties are similar to the properties of continuous functions. We denote by Scl(X)  the ring of all real valued strongly Ɵcl - continuous functions on a topological space X . It is proved that if the range of a strongly  Ɵcl- continuous function f  is a T1 - space, then f  is constant on quasi – components of its domain. Using this fact, we prove that for every topological space X , there is an ultra- Hausdorff  space Y  such that Scl(X) is isomorphic to C(Y).The behavior of these functions in relation to separation axioms is studied. We show that if f is a strongly Ɵcl - continuous function from a topological space X  to a T0 - space Y , then X  is  ultra-Hausdorff. Topological properties of direct and inverse image of spaces with certain topological properties under strongly Ɵcl - continuous functions are investigated. Among them, it is proved that the image of every cl-closure compact space under a strongly Ɵcl - continuous function is compact. Finally the properties of the graphs of strongly Ɵcl - continuous functions are discussed. It is proved that for every compact and Hausdorff space Y , strong Ɵcl - continuity of the function f from X to Y is equivalent to Ɵcl - closedness of  the graph of f relative to X .

In this paper we present some properties of set-norm exhaustive set multi functions and also of atoms and pseudo-atoms of set multi functions taking values in the family of non-empty subsets of a commutative semi group with unity.

In this paper we define the Soft set Product as a product of many soft sets and afterwards we extend it to the HyperSoft set. Similarly, the IndetermSoft Product is extended to the IndetermHyperSoft set. We also present several applications of the Soft set Product to Fuzzy (and fuzzy-extensions)  Soft set Product and to IndetermSoft set and IndetermHyperSoft set.