For each space and equivalence relation below, identify a familiar space homeomorphic to the quotient and prove your answer.
Let \(X=\mathbb R^2\) and define \[ (x_0,y_0)\sim(x_1,y_1) \quad\Longleftrightarrow\quad x_0+y_0^2=x_1+y_1^2. \]
Let \(X=\mathbb R^2\) and define \[ (x_0,y_0)\sim(x_1,y_1) \quad\Longleftrightarrow\quad x_0^2+y_0^2=x_1^2+y_1^2. \]
Let \(X=[-1,1]\) and let \(\sim\) be the smallest equivalence relation satisfying \(-p\sim p\) for every \(p\in X\).
Let \(X=\mathbb R\) and let \(\sim\) be the smallest equivalence relation satisfying \(x\sim x+n\) for every \(x\in\mathbb R\) and \(n\in\mathbb Z\).