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

• These two definitions are logically equivalent.

Also see

• The space $\R$ of real numbers is complete.

• Euclidean space $\R^n$ is complete.

• The metric space of rational numbers $\Q$ does not form a complete metric space.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$: Complete Metric Spaces