If S={x in R2: |x|=1} and S/~ is any quotient space, then is S always homeomorphic to S/~ ?

Does anyone know if this is true? Is there a counterexample?

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- November 2nd 2008, 09:10 PMziggychickquotient spaces
If S={x in R2: |x|=1} and S/~ is any quotient space, then is S always homeomorphic to S/~ ?

Does anyone know if this is true? Is there a counterexample? - November 3rd 2008, 02:19 PMclic-clac
No that's not true.

For example if you consider S and S/~ when ~ identifies two different points in S, then you get two non homeomorphic spaces (they have different fundamental groups, Z for S and Z*Z for S/~). - November 3rd 2008, 08:45 PMziggychick
i'm not sure what you mean by ~ identifies two points in S.

do you mean x~y iff x=-y?

or do you mean take two points in S and identify them with each other, then leave the rest the same?

are you familiar with the universal property of quotient spaces? that's where this question arrived from. - November 4th 2008, 10:45 AMclic-clac
*"do you mean take two points in S and identify them with each other, then leave the rest the same?"*

Yes.