Definition:Complete Metric Space

Definition

Definition 1

A metric space $M = \struct {A, d}$ is complete if and only if every Cauchy sequence is convergent.

Definition 2

A metric space $M = \struct {A, d}$ is complete if and only if the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty.

Equivalence of Definitions

These definitions are shown to be equivalent in Equivalence of Definitions of Complete Metric Space.

Also see

• Results about complete metric spaces can be found here.