5. Consider $X = [1, 2]^2$ with the relation $(x_1, y_1) \sim (x_2, y_2)$ if $x_1 y_2 = x_2 y_1$. Also define $g : X \to [1, 2]$ by the rule
$g(x, y) = \frac{3x}{x + y}$.
A. Write the equivalence class of $(x_0, y_0) \in X$.
B. Show that $g$ is constant on equivalence the classes of $\sim$.
C. Given that $g$ is a quotient map, show that $\{[(x, y)] : (x, y) \in X\}$ and $[1, 2]$ are homeomorphic. (Hint. Use Theorem 22.2 and #4)
Theorem 22.2
Let $p : X \to Y$ be a quotient map. Let $Z$ be a space and let $g : X \to Z$ be a map that is constant on each set $p^{-1}(\{y\})$, for $y \in Y$. Then $g$ induces a map $f : Y \to Z$ such that $f \circ p = g$. The induced map $f$ is continuous if and only if $g$ is continuous; $f$ is a quotient map if and only if $g$ is a quotient map.
4. Suppose $q : X \to Y$ is a quotient map and that it is one-to-one. Show that $q$ is a homeomorphism. (Hint. Since $q$ is injective, for any $A \subseteq X, q^{-1}(q(A)) = A$.)
