Measure, Integration & Real Analysis · Problem 6A.2
Banach Spaces — Sheldon Axler
Mathematics — Measure Theory · metric-spaces-finite-subsets-closed
Problem
Prove that every finite subset of a metric space is closed.Intuitive explanation
The key insight is that closed sets are exactly those containing all their limit points—and finite sets have none. In a metric space, a limit point of a set requires infinitely many distinct points from arbitrarily close to it. Since a finite set has only finitely many points total, it cannot accumulate anywhere, so it contains all its limit points vacuously. This makes the complement open (you can always find a small ball around any external point that avoids the finite set), proving the set itself is closed.
Intuitive explanation
The key insight is that closed sets are exactly those containing all their limit points—and finite sets have none. In a metric space, a limit point of a set requires infinitely many distinct points from arbitrarily close to it. Since a finite set has only finitely many points total, it cannot accumulate anywhere, so it contains all its limit points vacuously. This makes the complement open (you can always find a small ball around any external point that avoids the finite set), proving the set itself is closed.