Definition: A continuous mapping \( \displaystyle f : X \longrightarrow Y \) between two locally compact topological spaces is called
proper if the inverse image of every compact subset of \( \displaystyle Y \) under \( \displaystyle f \) is a compact subset of \( \displaystyle X \).
- 1. Give an example of a proper and of a non-proper mapping.
- 2. Prove that if \( \displaystyle f \) is a non-constant, proper, holomorphic mapping, then \( \displaystyle f \) is closed.
- 3. Prove that a non-constant, proper, entire function \( \displaystyle f \) is a non-constant polynomial.
- 4. Prove that if \( \displaystyle f \) is a non-constant, proper, holomorphic mapping and if \( \displaystyle A \) is a discrete subset of \( \displaystyle X \), then \( \displaystyle f \left( A \right) \) is a discrete subset of \( \displaystyle Y \).
