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?
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.