Putting Fuzzy Set Theory and Topology together is a key part of making decisions better in robots and autonomous systems. Fuzzy Set theory can version actual-world issues wherein trendy binary common sense won’t be able to, because it could deal with doubt and imprecision. That is specifically helpful in fields like sensor fusion, course planning, and manipulates structures, wherein the surroundings could make it difficult for computer systems to make selections on their own. Topology, however, offers us a solid manner to reflect on consideration on the shape and connections of space, which is very critical for planning motion, locating the exceptional paths, and making sure that the machine constantly acts the equal manner. By placing these two ideas together, we can make more complicated choice-making fashions that could better manage settings which can be changing, complicated, and unclear. This paper appears on the mathematical bases of both Fuzzy Set idea and Topology, looking at how they may be used one at a time and together in self-riding robots. We talk about actual-existence examples like robotic tracking, multi-robot teamwork, and self-sufficient vehicle manipulate to expose how these theories help make systems greater reliable, flexible, and excellent at making decisions. The results are useful for planning the future growth of smart systems that can run on their own.