Basic properties of open sets

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

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

Proof sketch. (1)–(2) follow directly from the definition. For (3), if $x$ lies in a union, it lies in one member open set and hence has a ball contained in that member and thus in the union. For (4), intersect the finitely many balls given by each open set at a point: taking the minimum radius keeps the ball inside all sets.