1. ## 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?

2. 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/~).

3. 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.

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