I don’t want to say topology is taught wrong, but I do feel like it’s popularized in the wrong way. You’ll hear people say, “Oh, topologists, they are very interested in surfaces that you can bend and stretch but you can’t cut or glue.” Are they? Why?… The short answer is, [that’s not really what they are interested in]. It’s not as if someone was sitting there thinking “I wonder what the properties of clay are if I add some arbitrary rules about how I can’t cut it or glue it.” Instead, there are a ton of pieces of math that can be equivalent to these very general structures that are like geometry except that you don’t have exact distances, you just want to maintain a notion of “closeness”. And once you get into those general structures, constructing mapping between them translates into non-trivial facts about other parts of math. I don’t think that is popularized enough. I don’t even think it’s emphasized enough when you’re starting to take a topology class.