I am a sophomore and still taking calculus 2 and 3. However, I asked several questions in class but the professor always answers me: you need to take a higher level of mathematics, topology in specific, to be able to answer these questions.
For example, when we were taking how to derive and integrate functions with multiple variables, I noticed that both differentiation and integration are generally expressed in terms of limits. In fact, this was a shock for me because in high school I used to treat derivatives, integrals, and limits such as every one of them have its own rules and I thought that they are somehow independent. It was the first time I see their proofs and understand what they really mean. So I've just wondered and asked if there is anything more general that can define limits. The professor answered: topology.
Topology again! It was the third time this semester I received the same mysterious answer.
Therefore, I decided to search and read some books related to topology. However, most explanations and books require a higher level of mathematics than I have.
Can anyone explain the general idea of topology and recommend some simple books?
Best Answer
I would suggest (at least temporarily) avoiding Topology, and instead looking for a somewhat proof-oriented textbook on Real Analysis. One example is "Calculus" (volumes I and II), 2nd Ed. (Tom Apostol).
However, this may not be the right book for you. You have to find a proof-oriented Real Analysis book, with a lot of exercises, that dovetails with your interests, Math education, and Math ability. You want the book to (only moderately) stretch your intution.
Depending on the nature of your questions, you may be able to resolve all of the questions that you are curious about via the right Real Analysis book. If so, this would allow you to avoid a deeper study of Topology. This avoidance is important if Topology does not happen to be something that you are independently interested in.