You're right! That makes so much more sense now! The disjoint open sets would then just be the periodic repeats of the arcs. I need to read more carefully. Thank you so much - you've made my day! :)
I'm probably interpreting this wrongly but doesn't it say:
disjoint union of open sets each of which is mapped homeomorphically
onto U _alpha by p.
Thanks for being so patient.
But p was defined to be p(s)=(cos 2*pi*s, sin 2*pi*s). Doesn't that mean that the preimage of a cover in S1 must cover an interval in R? Also, it says that each p-1(U_alpha) was mapped homeomorphically to U_alpha by p. Doesn't that mean that connectedness must be preserved by the inverse...
This is from Hatcher's Algebraic Topology, page 30.
I thought that the circle, S1 is path connected. How then can it be decomposed into the disjoint union of open sets? Furthermore, how can two disjoint open arcs in S1 be the have S1 as their union? What happened to the boundary of the two...
Maybe you could adopt the asian method and just make the students do more and hope it works. XD
The australian maths syllabus is a year behind malaysian and singaporean syllabi and their students are no more competent at what they learn either. The students in the asian countries do more...
It's true. For chinese students (not just China), becoming a well-paid professional is something that has been ingrained in our culture. Becoming a doctor is at the very top. Probably to do with some confucian value. Accountants and engineers are also pretty well-respected. Law is also...
Tao's Analysis starts from the very foundations and in doing so, forces you to develop proof skills in a very intuitive environment (natural numbers, integers, etc.). I've fallen in love with Pugh's book. And Munkres is great. In fact, it's a better first book than baby Rudin since it does...
I should hope that anyone who's doing the study group with us has at least a basic understanding of these topics. Anyone who isn't is not really prepared to study anything else pure or applied.
And yes, do pm me. We can adjust the syllabus as we go along according to interest. Likewise with...
And yes, those are applied topics. Almost all the topics I have mentioned are applied maths topics. If you don't believe me, look up: Boas, Stroud, Kreyzieg, Arfken...
I'm thinking that you should start on multivariable calculus. It's a good area where derivations and proofs go together very well. Start with vector fields and then move on to vector bundles and tensors perhaps? That's pretty important for physicists and some engineers. Or you could go the...