Some applications on homotopy theory in scientific fields

We introduced the topological fundamental group and presented some interesting basic properties of the notion. Also we extend the notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more about the topology of the topological homotopy groups order to find necessary and sufficient conditions for which the topology is discrete. We followed the analytical induction mathematical method and we found that studying homology groups may be more than cohomology fundamental groups.