Definition:Complete Metric Space

Definition
A metric space $\left({X, d}\right)$ is complete if every Cauchy sequence is convergent.

Alternative Definition
A metric space $\left({X, d}\right)$ is complete iff the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty.

Equivalence of Definitions

 * These two definitions are logically equivalent.

Also see

 * The space $\R$ of real numbers is complete.


 * Euclidean space $\R^n$ is complete.


 * The metric space of rational numbers $\Q$ does not form a complete metric space.