The main purpose of this paper, is to introduce a topological space , which is induced by reflexive graph and tolerance graph , such that  may be infinite. Furthermore, we offered some properties of  such as connectedness, compactness, Lindelöf and separate properties. We also study the concept of approximation spaces and get the sufficient and necessary condition that topological space is approximation spaces.

This paper focuses on developing a strategy to represent the -connected ominoes using an abacus. We use the idea of -connected ominoes with respect to a frame in modelling nested chain abacus. Then, we formulate and prove the unique connected partition for any -connected ominoes. Next, the topological structure of nested chain abacus is presented.

In this paper, we introduced some new definitions on P-compact topological ring and PL-compact topological ring for the compactification in topological space and rings, we obtain some results related to P-compact and P-L compact topological ring.

Let  be a prime ring,  be a non-zero ideal of  and   be automorphism on. A mapping  is called a multiplicative (generalized)  reverse derivation if  where  is any map (not necessarily additive). In this paper, we proved the commutativity of a prime ring R admitting a multiplicative (generalized)  reverse derivation  satisfying any one of the properties:

 for all x, y  

In this present paper, we obtain some differential subordination and superordination results, by using generalized operators for certain subclass of analytic functions in the open unit disk. Also, we derive some sandwich results.

    The paper starts with the main properties of the class of soft somewhere dense open functions and follows their connections with other types of soft open functions. Then preimages of soft sets with Baire property and images of soft Baire spaces under certain classes of soft functions are discussed. Some examples are presented that support the obtained results. Further properties of somewhere dense open functions related to different types of soft functions are found under some soft topological properties.

In this paper the definition of fuzzy normed space is recalled and its basic properties. Then the definition of fuzzy compact operator from fuzzy normed space into another fuzzy normed space is introduced after that the proof of an operator is fuzzy compact if and only if the image of any fuzzy bounded sequence contains a convergent subsequence is given. At this point the basic properties of the vector space FC(V,U)of all fuzzy compact linear operators are investigated such as when U is complete and the sequence ( ) of fuzzy compact operators converges to an operator T then T must be fuzzy compact. Furthermore we see that when T is a fuzzy compact operator and S is a fuzzy bounded operator then the composition TS and ST are fuzzy compact

         Let  be any group with identity element (e) . A subgroup intersection graph of  a subset  is the Graph with V ( ) =  - e and two separate peaks c and d contiguous for c and d if and only if      , Where  is a Periodic subset of resulting from  . We find some topological indicators in this paper and Multi-border (Hosoya and Schultz) of   , where    ,  is aprime number.

  The purpose of this paper is to study a new types of compactness in bitopological spaces. We shall introduce the concepts of  L- compactness.