# Search results

1. ### 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. ### Post your spring schedule here!

So, what courses are you going to take spring 2014? Post it here!
3. ### 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. ### A physical description of some GR concepts

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