Irresolute Homotopy and Covering Theory in Irresolute Topological Groups

In this paper, we explore certain properties related to connectedness and introduce the definition of irresolute paths. Subsequently, we define the concepts of semi-path connectedness, locally semi-path connectedness, and semi-locally s-simply connected spaces. Additionally, we introduce the concept of irresolute homotopy and reconstruct the fundamental group based on this framework. Furthermore, we prove that the structure of an irresolute topological group with a universal irresolute covering can be lifted to its irresolute covering space.