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...
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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...
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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...
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##...
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?
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...
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...
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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...
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...
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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!
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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...