Exercise 1, § 10

a) Give an example of continuous mappings $f : X \to Y,\ g : Y \to Z$ such that $g \circ f$ is proper, $f$ is not surjective and $g$ is not proper. b) Give an example of two proper mappings $f : X \to Y$ and $g : X \to Z$ where $X, Y, Z$ are quasi-compact, such that $x \to (f(x), g(x))$ is not a proper mapping of $X$ into $Y \times Z$.