Measure, Integration & Real Analysis · Problem 6A.5
Banach Spaces — Sheldon Axler
Mathematics — Measure Theory · metric-spaces-closed-sets
Problem
Suppose is a metric space. (a) Prove that the intersection of each collection of closed subsets of is a closed subset of . (b) Prove that the union of each finite collection of closed subsets of is a closed subset of .Intuitive explanation
Closed sets in metric spaces are defined by their complements: a set is closed if and only if its complement is open. The key insight is to use De Morgan's laws to flip between unions and intersections of complements, then apply the fundamental properties of open sets (arbitrary unions are open, finite intersections are open). This transforms a statement about closed sets into one about their complements, which is often easier to verify.
Intuitive explanation
Closed sets in metric spaces are defined by their complements: a set is closed if and only if its complement is open. The key insight is to use De Morgan's laws to flip between unions and intersections of complements, then apply the fundamental properties of open sets (arbitrary unions are open, finite intersections are open). This transforms a statement about closed sets into one about their complements, which is often easier to verify.