N-dimensional Unit Ball, N-dimensional Ellipsoid and Topology Problems
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises. –James R. Munkres
It is said that mathematics is not a spectator sport. It is a sport which is played by the audience. The way the audience play it is by doing mathematics themselves. A crucial part of mathematical learning is solving problems. If you can solve the problems, it shows you aren't just talking the talk but you are walking the walk. When it comes to solving problems, a student should try to lay hands on whatever problems s/he can. Problems given in books at the end of a chapter or topic is a great source. Solving those problems helps not only in testing one's understanding but also increases one's confidence in solving problems. Solving problems helps in preparing for research too. By solving tough problems that are usually given near the end of problem sets, one can simulate a research experience. Since a second-year undergrad has very little knowledge to encounter a research problem, my guide suggested that I should focus more on increasing my understanding of the concepts rather than attack research problems directly. So, he proposed that I solve as many problems as I can. In this document, I will start by providing the solution for n-dimensional unit ball problem and n-dimensional ellipsoid problem, followed by problems from Topology. The unit ball problem was proposed to me by my guide. He had solved such a problem in one of his papers in the reputed journal 'Resonance'. Most parts of the current document consist of the solutions of end topic problems from Topology by Munkres. In the end, I have also presented the solutions for a few problems from the first chapter of "principles of mathematical analysis" by Walter Rudin. I could have given the solutions to more problems but I feared that would far exceed the required length of the document.
Keywords: Topology, mathematical analysis, n-dimensional unit ball