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/~).
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.