# Definition:Complete Metric Space

## Definition

### Definition 1

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

### Definition 2

A metric space $M = \left({A, d}\right)$ 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.