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  1. R136a1

    Spring with 2 masses free fall

    Homework Statement The Attempt at a Solution The reference to exercise 1.7 is not essential. The only part exercise 1.7 that is relevant to this exercise is the following: if a long narrow tube is drilled between antipodal points on a sphere of uniform mass density and two identical...
  2. R136a1

    Post your spring schedule here!

    So, what courses are you going to take spring 2014? Post it here!
  3. R136a1

    Naber's Topology, geometry and gauge fields and similar books

    Hello, This thread is about the two books by Naber: https://www.amazon.com/dp/1461426820/?tag=pfamazon01-20&tag=pfamazon01-20 https://www.amazon.com/dp/0387989471/?tag=pfamazon01-20&tag=pfamazon01-20 The topics in this book seem excellent. They are standard mathematical topics such as...
  4. R136a1

    A physical description of some GR concepts

    Can someone explain what the expansion, rotation, and shear of a time-like congruence are physically?
  5. R136a1

    Abelian fundamental groups

    Hello everybody! So, I've learned that in a path connected space, all fundamental groups are isomorphic. Indeed, if ##\gamma## is a path from ##x## to ##y##, then we have an isomorphism of groups given by \Phi_\gamma : \pi_1(X,x)\rightarrow \pi_1(X,y): [f]\rightarrow [\overline{\gamma}]\cdot...
  6. R136a1

    Difficulty checking group axioms

    Let ##G## be a set equipped with a binary associative operation ##\cdot##. In both of the following situations, we have a group: 1) ##G## is not empty, and for all ##a,b\in G##, there exists an ##x,y\in G## such that ##bx=a## and ##yb=a##. 2) There exists a special element ##e\in G##...
  7. R136a1

    About closed sets in the plane

    I'm wondering if the following is true: Every closed subset of ##\mathbb{R}^2## is the boundary of some set of ##\mathbb{R}^2##. It seems false to me, does anybody know a good counterexample?
  8. R136a1

    Homotopic maps on a sphere

    If I take two arbitrary continuous maps ##f,g:S^n\rightarrow S^n## such that ##f(x) \neq -g(x)## for any ##x\in S^n##, then ##f## and ##g## are homotopic. How do I show this result? I really don't see how to use the condition that ##f## and ##g## never occupy two antipodal points. Any hint...
  9. R136a1

    Some questions about topological groups

    So, I have a topological group ##G##. This means that the functions m:G\times G\rightarrow G:(x,y)\rightarrow xy and i:G\rightarrow G:x\rightarrow x^{-1} are continuous. I have a couple of questions that seem mysterious to me. Let's start with this: I've seen a statement...
  10. R136a1

    Continuity of the maximum

    Hello everybody! Given a topological space ##X## and two functions ##f,g:X\rightarrow \mathbb{R}##, it is rather easy to prove that ##x\rightarrow \max\{f(x),g(x)\}## is continuous. I wonder if this also holds for infinitely many functions. Of course, the maximum doesn't need to exist, so we...
  11. R136a1

    Box topology does not preserve first countable

    So, in the topology text I'm reading is mentioned that if each ##X_n## is first countable, then ##\prod_{n\in \mathbb{N}} X_n## is first countable as well under the product topology. And then it says that this does not need to be true for the box topology. But there is no justification at all...
  12. R136a1

    Hausdorff spaces

    Hello everybody! It is known that in Hausdorff spaces that every sequence converges to at most one point. I'm curious whether this characterizes Hausdorff spaces. If in a space, every sequence converges to at most one point, can one deduce Hausdorff? Thanks in advance!
  13. R136a1

    Second Countable spaces

    Hello everybody! I hope I'm posting in the correct forum. Apparently, I can post questions from grad books in this forum, so I decided to post here! The topology book I'm using asks me to prove that if ##X## is a second countable topological space, if ##x\in \overline{A}##, then there...
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