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

 * Equivalence of Definitions of Complete Metric Space

Also see

 * Real Number Line is Complete Metric Space
 * Euclidean Space is Complete Metric Space


 * The Space of Rational Numbers is Not Complete