Basic properties of open sets - Imported Knowlpedia Content

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

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

Remarks

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.