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Just another ordinary day with another (great) ordinary mathy conversation. A friend and I got to ask this:

Problem. Prove that $B(0,r) \setminus \{0 \} \subseteq \mathbb{R}^n$ is not homeomorphic to open balls for $r > 0$.

I have not yet had a topology course and my friend has just started a topology course, so we tried to find a solution as elementary as possible.

It seems easy enough, though we struggled...

  • It suffices to prove that it's not homeomorphic to $\mathbb{R}^n$.
  • Most elementary topological invariants don't work.

The best we came up with, is to compute the fundamental group and take generalizations of the fundamental group for $n > 2$. That should probably work.

So here's my question: Is there a different way to prove this result?

Qi Zhu
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    Do you count using homology as a "different way"?, – Yuval May 01 '19 at 17:24
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    I unfortunately do not yet know what that is. However, I do thank you for throwing in homology, that does help! – Qi Zhu May 01 '19 at 17:27
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    For $n=2$ you can use a topological invariant, see the very nice proof here. Using $B(0,R)-0\simeq (0,1)\times \mathbb{S}^{n-1}$ you should be able to adapt the proof for any $n$. – Adam Chalumeau May 01 '19 at 17:35

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As done here you can use the following topological invariant:

For every compact subset $K$ of $X$ there is a compact $K^\prime\subset X$ containing $K$ such that $X-K^\prime$ is connected.

$\mathbb{R}^n$ has this property, but $\mathbb{R}^n-\{0\}$ (or $B(0,r)-\{0\}$) doesn't.

Adam Chalumeau
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    This should be "$X - K'$ is connected". – Mark Kamsma May 01 '19 at 17:55
  • @MarkKamsma thanks I edited ! – Adam Chalumeau May 01 '19 at 17:56
  • Nice one. Do you have any motivation on the invariant? (Why should it make sense that it's a topological invariant, etc.) – Qi Zhu May 01 '19 at 19:51
  • Well it's a property that involves compacts, inclusion, connectedness... so "it has to be an invariant !" (you can give a real proof in this case). I can't really give a motivation for this invariant, I don't know how it was found by the person in the linked post. It's the kind of think you read in a book and transmit to somebody else etc... – Adam Chalumeau May 01 '19 at 20:20
  • Well yeah, that‘s the only thing I can think of without spending too much time on it, as well. (Also, it has to do with connectedness, so possibly it will identify holes, hence be useful to this problem.) I‘m interested in how the person found it. Yes, there are things you read in books that are miraculous but my goal is always to try to understand how one would come up with it (someone had to come up with it first!). Anyway, thanks for your reply! – Qi Zhu May 02 '19 at 06:07
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    @Kezer I found that this property has a name, you can look at this – Adam Chalumeau May 11 '19 at 13:57
  • Wow, cool, thank you a lot! – Qi Zhu May 12 '19 at 13:09