The Role of Group Action

In this chapter, we shortly present some prerequisites for the book concerning mathematical structures in calculus and geometry. We introduce and discuss in detail, especially for the benefit of physicists, the notion of quotient spaces in connection with equivalence relations. The last two sections deal with group actions, which are the mathematical face of what we meet as symmetries in physics. This chapter could be considered as appendix, but we set it at the beginning of the book to point out its significance. For physicists, it can be skipped on the first reading.