Complete metric space

Definition

A metric space $(X, d)$ is complete when each Cauchy sequence in $X$ converges to a point of $X$.

Remarks

The real line $\mathbb{R}$ with the usual metric is complete. So is $\mathbb{R}^n$ with the Euclidean metric. The rationals $\mathbb{Q}$ and the open interval $(0, 1)$ are not complete. In $(0, 1)$, the sequence $1/n$ is Cauchy but has no limit in $(0, 1)$. Completeness depends on the metric, not only on the topology: $(0, 1)$ is homeomorphic to $\mathbb{R}$, but only $\mathbb{R}$ is complete with the usual metric. Each metric space has a completion.

Explanations

L1 human

A complete space has no holes: each sequence that looks convergent really has a limit in the space. The rationals have holes, for example at $\sqrt{2}$, and the reals fill them.