Knowlpedia

Basic properties of closed sets

Intersections of closed sets are closed; finite unions of closed sets are closed

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

  1. ∅\emptyset is closed.
  2. XX is closed.
  3. The intersection of any collection of is closed.
  4. The union of finitely many closed sets is closed.
Remarks

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