Basic properties of closed sets

Proposition. Let $(X,d)$ be a metric space. Then:

1. $\emptyset$ is closed.
2. $X$ is closed.
3. The intersection of any collection of closed sets is closed.
4. The union of finitely many closed sets is closed.

Proof sketch. Use complements and the corresponding results for open sets: complements turn arbitrary intersections into arbitrary unions and finite unions into finite intersections.