Definition:Complete Metric Space

This page is about Complete Metric Space. For other uses, see Complete.

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.

Also known as

A complete metric space is also known as a complete space.

Also see

• Equivalence of Definitions of Complete Metric Space

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

• Rational Number Space is not Complete Metric Space

• Results about complete metric spaces can be found here.

Sources

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• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $3.12$
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $4$: Complete Normed Spaces