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Math Help - quotient spaces

  1. #1
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    quotient 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?
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  2. #2
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    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/~).
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  3. #3
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    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.
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  4. #4
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    "do you mean take two points in S and identify them with each other, then leave the rest the same?"
    Yes.
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