Definition:Complete Metric Space

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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.


Sources