We introduce a topology on the set of vertices of a directed graph and we call the topological space as pathless directed topological space. We study relation between the relative topologies and pathless directed topological spaces of E-generated subdirected graphs. Then, we study connectedness, isomorphic and homeomorphic properties in digraphs and pathless directed topological spaces. Moreover, we apply our results to blood circulation process in human heart and disprove Shokry and Aly [M. Shokry and R. E. Aly, Topological properties on graph vs medical application in human heart, Int. J. Appl. Math., 15 (2013), 1103-1109], Nada et al. [S. Nada, A. E. F. El Atik and M. Atef, New types of topological structures via graphs, Math. Method. Appl. Sci., 41 (2018), 5801-5810] and Nawar et al. [A. S. Nawar and A. E. F. A. El-Atik, A model of a human heart via graph nano topological spaces, Int. J. Biomath., 12 (2019), p.1950006]. We show that pathless directed topology is accurately describing the circulation of blood in the heart of human body.

Our main objective of this work was to study rough approximations of spherical fuzzy ideals by using soft relations that are free from all the complexities that are faced by many scientists. Furthermore, lower and upper approximations of spherical fuzzy subsemigroups, spherical fuzzy left (right) ideals, spherical fuzzy interior ideals, and spherical fuzzy bi-ideals of semigroups were studied using soft relations. Mainly, we proved, for a spherical fuzzy ideal of the universe, that upper and lower approximations are spherical fuzzy soft ideals but the converse may not hold, as shown by examples. Compatible relations and complete relations are needed for upper approximations and lower approximations, respectively. Also, using examples, we showed that the conditions of complete relations were necessary for lower approximations. Last, a comparison study and conclusions of the introduced technique are given, demonstrating how our work is superior and efficient in contrast to other techniques.