According to the above definition, the most basic concept is:
1. "Topological space" refers to defining geometry in a total set-what is a point, what is a distance, what is a dimension, and so on. Its definition is basically understandable for high school students, but it is too abstract to really understand;
2. "homeomorphism". This definition is basically understandable for high school students. However, it is impossible to directly judge whether two topological spaces are homeomorphic, so a series of "invariants"-Euler characteristics, homology groups, cohomology groups, etc. are introduced into each topological space. If one of the Euler characteristics/homology groups/cohomology groups in two topological spaces is different, it can't be homeomorphism (that's how Mobius circle you mentioned differs from ordinary circle in topology because their homology groups are different). Among these invariants, Euler features are integers, and the definition is not complicated, so it should be no problem for high school students to understand; But homology group/cohomology group is not a number but a "group", and the concept of group itself can only be learned in universities, so it is impossible for high school students to understand the definition of homology group/cohomology group;
3. Homotopy. The same as above, but the invariant is changed from homology group to homotopy group, and its definition is also incomprehensible to senior high school students;
4. "Vector Bundle", "Fiber Bundle" and their "Chen Shengren Class". These are impossible for even junior students in the department of mathematics to understand;
etc
The conclusion is: you can study. Ideally, you have a little knowledge-ideally, you are at a loss, so unless you are really fascinated, you can't sleep at night, or you should spend your time on something more suitable for senior high school students.