(a) Deﬁne an equivalence relation on the plane $\displaystyle X=\mathbb{R}^2$ as follows:

$\displaystyle x_0 \times y_0 $ ~ $\displaystyle x_1 \times y_1$ if $\displaystyle x_0 + y_0^2=x_1 + y_1^2.$

Let $\displaystyle X^*$ be the corresponding quotient space. It is homeomorphic to a familiar space; what is it? [Hint: Set $\displaystyle g(x \times y)=x+y^2.$]

(b) Repeat (a) for the equivalence relation

$\displaystyle x_0 \times y_0 $ ~ $\displaystyle x_1 \times y_1$ if $\displaystyle x_0^2+y_0^2=x_1^2+y_1^2.$

[this is Munkres §22.4]