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Abstract Topology is one of the most significant branches of mathematics, which plays a crucial part in real life applications. In this thesis, we use a graph theory and rough set theory to generate new forms of nano topological structures from any simple directed graph. Moreover, a digraph model of a human blood circulation is introduced and nano topological spaces are generated. Some definitions of separation axioms depended on topological structures of graphs are studied in both the digraph of the heart and any other simple directed graph. Simplicial complexes also are played an important role in an algebraic topology. They are geometric objects that can be created and used in real-world situations. A universal set of any simplicial complexes are introduced as a set of vertices, edges, faces, tetrahedrons, and so on. A betweenness relation on any simplicial complex is defined and some of its topologies are studied. Matroids are offered as abstract extensions to the linear independency in vector spaces and cycles in graphs. They have strong theoretical foundations and a wide range of applications. In this thesis, we make connections between simplicial complexes, matroid theory and a rough set theory to generate new types of matroids called simplicial matroids. Some of these simplicial matroids are generated from any simplicial complex by defining special kind of matrices on it. Some of them are generated by defining special kinds of relations on the simplicial complex such as equivalence relation and order relation. Moreover, we study some of matroidal characterizations of these simplicial matroids. |