Concept of Prepaths, Path Cycles Spaces and Bond Spaces

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Abstract

We demonstrate that all paths, and the topological speculations of cycles, are topologized graphs. We utilize weak normality to investigate connections between the topologies on the vertex set and the entire space. We utilize minimization and frail typicality to demonstrate the presence of our analogs for negligible traversing sets and essential cycles. In this system, we sum up theorems from finite graph theory to an expansive class of topological structures, including the actualities that crucial cycles are a reason for the cycle space, and the orthogonality between bond spaces and cycle spaces. We demonstrate this can be refined in a setup where the arrangement of edges of a cycle has a free relationship with the cycle itself. Things being what they are, in our model, weak normality prohibits a few pathologies, including one distinguished, in an altogether different approach which tends to similar issues.