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I am new to quotient spaces and was given this in class. I really have no idea how to solve. I tried one approach that my teacher said to be incorrect so I'd really appreciate the help on this.

I am given $$S^1= \{x \in \mathbb{R}^2 : \|x\|=1 \}$$ i.e., the unit circle in $\mathbb{R}^2 $, and I am asked to show the quotient space $ S^1 / {\sim}$, where $\sim$ is the equivalence relation $x\sim{-x}$, is homeomorphic to $ S^1 $.

My attempt was to look at the quotient space as the set of points of the unit circle with only one of $(1,0)$ and $(0,1)$, which I now realize is incorrect as I know quotient space means gluing as opposed to omitting points. I am stumped as I cannot really figure out how to show an explicit homeomorphism between the quotient space and the unit circle in $\mathbb{R}^2$, or if I am even on the right track (maybe I need to use a theorem or something without a constructive proof). I really would appreciate the help on this. Thanks all.


marked as duplicate by Zev Chonoles, Adam Hughes, drhab, Alex S, user147263 Jul 8 '15 at 18:01

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    $\begingroup$ I hope you find my gif animations at the linked question helpful :) $\endgroup$ – Zev Chonoles Jul 8 '15 at 5:31
  • $\begingroup$ Thanks @ZevChonoles appreciate this but still I feel yours does not answer my question I do know they are homeomorphic but in this question I am asked to prove the homeomorphism. I couldn't find a proof in your answer for them being actually homeomorphic $\endgroup$ – kroner Jul 8 '15 at 5:34
  • $\begingroup$ There are other answers on that thread with actual formal arguments, but the animations capture the intuition behind it. $\endgroup$ – Zev Chonoles Jul 8 '15 at 5:35
  • $\begingroup$ Thanks @ZevChonoles will check $\endgroup$ – kroner Jul 8 '15 at 5:38
  • $\begingroup$ @ZevChonoles And after spending so much effort on making illistrations, you've left the title of that question untouched... (edited now). $\endgroup$ – user147263 Jul 8 '15 at 18:01

Label each point on the circle by the angle $\theta\in [0,2\pi)$. Let $f:S/{\sim}\rightarrow S$ be defined by $f(\theta)=2\theta$. You should be able to prove this is well-defined on $S/{\sim}$, and is in fact a bijection between the two spaces. Then to prove it's a homeomorphism, you just need to show that the image/preimage of any open set is open.


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