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.

In this paper we show that if ? Xi is monotonically T2-space then each Xi is monotonically T2-space, too. Moreover, we show that if ? Xi is monotonically normal space then each Xi is monotonically normal space, too. Among these results we give a new proof to show that the monotonically T2-space property and monotonically normal space property are hereditary property and topologically property and give an example of T2-space but not monotonically T2-space.

In the present paper, we have introduced some new definitions On D- compact topological group and D-L. compact topological group for the compactification in topological spaces and groups, we obtain some results related to D- compact topological group and D-L. compact topological group.

     In this paper, we study a new concept of fuzzy sub-module, called  fuzzy socle semi-prime sub-module that is a generalization the concept of semi-prime fuzzy sub-module and fuzzy of approximately semi-prime sub-module in the ordinary sense.  This leads us to introduce level property which studies the relation between the ordinary and fuzzy sense of approximately semi-prime sub-module. Also, some of its characteristics and notions such as the intersection, image and external direct sum of fuzzy socle semi-prime sub-modules are introduced. Furthermore, the relation between the fuzzy socle semi-prime sub-module and other types of fuzzy sub-module presented.

The research aims to conduct a comparative study between the events (800) m and (1500) m track in the types of muscular strength and to identify the differences between them among Iraqi sports club players. The sample represented the players participating in the Iraqi Athletics Championship for the period between (02/11/2023) and (04 /11/2023), and their number was (16) players (8) for each event, as the selection was made intentionally.  The researchers used the descriptive approach to achieve the goal of the research and used the statistical package (SPSS) to process the data statistically. According to the results collected, it was found that there was superiority among the intermediate track players (800) m in explosive power and the s

Relation on a set is a simple mathematical model to which many real-life data can be connected. A binary relation  on a set  can always be represented by a digraph. Topology on a set  can be generated by binary relations on the set . In this direction, the study will consider different classical categories of topological spaces whose topology is defined by the binary relations adjacency and reachability on the vertex set of a directed graph. This paper analyses some properties of these topologies and studies the properties of closure and interior of the vertex set of subgraphs of a digraph. Further, some applications of topology generated by digraphs in the study of biological systems are cited.

     Let R be a commutative ring with unity and an R-submodule N is called semimaximal if and only if

 the sufficient conditions of F-submodules to be semimaximal .Also the concepts of (simple , semisimple) F- submodules and quotient F- modules are  introduced and given some  properties .

         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 define fuzzy subspaces for fuzzy space of orderings and we prove some results about this definition in which it leads to a lot of new results on fuzzy space of orderings. Also we define the sum and product over such spaces such that: If f = < a1,…,an > and g = < b1,…bm>, their sum and product are f + g = < a1…,an, b1, …, bm> and f × g =. for all a1,…,an,b1,…,bm ? G

Increased diseases and obesity currently due to increased production and excessive consumption of foods manufactured from non-food sweeteners without attention to the risk of consuming those additional high calories due to consuming these refreshing products such as juices and other various drinks, especially in the summer season by most segments of Iraqi society, especially workers, children and school students the aim of this study. Therefore, the study designed to replace sucrose with 0.03, 0.04 and 0.05% of each of the white stevia crystals and milled dry stevia leaves in the laboratory manufacture of juices and its effect on the general and sensory characteristics and the extent of their acceptability among the specialized r