Describe the quotient space $Y=S'/((1,0)\sim(0,-1)\sim(0,1)\sim(-1,0))$

$S'$ be the unit circle in $R^2$

I just simply cant visualise this space

also how do i check if its connected or compact?

any help greatly appreciated.

I am following Munkres topology and no examples like this or anything similar, completely stuck.

  • 1
    $\begingroup$ Could you define $S'$? Or is this is a typo, and is it supposed to be the 1-sphere $S^1$? $\endgroup$ – B. Pasternak May 25 '16 at 19:32
  • $\begingroup$ In that case, you have just identified these 4 points on the circle..can you try and imagine what this will look like? Try doing it point per point: pick for example (1,0) and identify (0,1) with this point, and so on. What do you get? $\endgroup$ – B. Pasternak May 25 '16 at 19:45

You have taken the circle and decided to consider four points interchangeable. It doesn't matter what four points they are. You can think of pushing each one in to the center so they are all $(0,0)$ and you have a four petal flower. Each petal comes from one quadrant of the circle. A sketch:

enter image description here

  • $\begingroup$ i understand these 4 points are equal to origin however i don't get why they will become petals?. And thanks for the answer by the way. $\endgroup$ – MRK May 25 '16 at 19:48
  • $\begingroup$ Draw this, then it will make sense. Ross, why do I get 3 petals? What do I miss? $\endgroup$ – B. Pasternak May 25 '16 at 19:48
  • $\begingroup$ also am i right in saying this is both compact and connected $\endgroup$ – MRK May 25 '16 at 19:48
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    $\begingroup$ The arc between each pair of points becomes one petal, so there are four. One comes from each quadrant of the circle. Yes, it is compact and connected. $\endgroup$ – Ross Millikan May 25 '16 at 20:13
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    $\begingroup$ @B.Pasternak: I suspect you are missing the petal that comes between the last point you identify and the first. You got the three between the first/second, second/third, and third/fourth. $\endgroup$ – Ross Millikan May 25 '16 at 20:17

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