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Prove $S^3$ and $\mathbb{R}^3$ are not homeomorphic.

I've encountered this question on a PhD exam in topology. This is at a level where we are expected to understand cohomology already, so there are already a lot of obvious one line proofs I could give (e.g. they don't have the same homology groups). But this appears among the "give a detailed answer" questions, as opposed to the more computational questions in the latter half. (The heading says to show all work and support all statements to the best of my ability.)

So, I am confused about the level of detail I would have to include here. Is there an obvious choice for how to prove this directly without relying on any one liners that assume higher level stuff?

I realize this question is a little opinion based, but maybe the answer will be unambiguous to those with more experience in topology. How would you answer this question?

hardmath
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Ng-ettingit
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  • You have stated the problem to be answered only in the title. Please include the problem statement in the body of the Question as well, usually with additional details (that may not be needed or entailed here). – hardmath Aug 12 '16 at 00:47
  • You get a close vote from me: you (more or less) know the answer and are concerned with how to get the exam points - that's not in scope for MSE. – Rob Arthan Aug 12 '16 at 00:56
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    Pretty annoying way to state the expectations... Is it OK to just say $S^3$ is compact? Do you need to say it's closed and bounded? (Do you need to prove it is closed and bounded?) Is that enough, or do you need to prove that closed and bounded implies compact? Where do you draw the line? If I was the grader, paf's answer would be a perfect score. –  Aug 12 '16 at 00:57
  • I agree @mathguy, and if you'll excuse the whine, the main reason I have trouble studying for this exam is that many of the other questions are stated with this sort of ambiguity. – Ng-ettingit Aug 12 '16 at 01:35
  • @RobArthan That is not true, and I resent the implication that I am grade grabbing. I am asking whether those more experienced than me understand (what I perceive to be) an ambiguously worded question. Even setting aside from the material content of the question, interpretation of mathematical writing is well within the scope of MSE. – Ng-ettingit Aug 12 '16 at 01:39
  • Usually you want to show as much detail as you can for important exams in graduate school. Try to give a proof that someone outside of topology could read and be able to determine its correctness. – Merkh Aug 12 '16 at 01:42
  • @Merkh - that is a good way to think about it. With that said, anyone who cannot understand paf's solution, exactly as written, doesn't belong in graduate school (not even as a student, much less grading others' exams). (Which is not to say they don't exist, in spades.) –  Aug 12 '16 at 02:02

2 Answers2

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$S^3$ is compact whereas $\Bbb R^3$ is not. Thus they cannot be isomorphic.

paf
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3

$S^3$ is compact, while $\mathbb{R}^3$ is not. Since any continuous function $f:S^3\rightarrow \mathbb{R}^3$ maps compact subsets of $S^3$ to compact subsets of $\mathbb{R}^3$, it can't be surjective (or else $f(S^3)=\mathbb{R}^3$ is also compact).

Sachchidanand Prasad
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