---
tag: 0G4J7Z7WPJ
kind: definition
name: Completion of a metric space
msc: ["54E50"]
defines: 06RW4RKWH7
uses: [08EWT7J0SC, 0878DN3WCS, 0D198GPMTJ]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_metric_space
    locator: Completion
---

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
A completion of $X$ is a pair $(\hat{X}, \iota)$ of a [[complete metric space|08EWT7J0SC]] $\hat{X}$ and an [[isometry|0878DN3WCS]] $\iota \colon X \to \hat{X}$ with [[dense|0D198GPMTJ]] image $\iota(X)$.

## Remarks

Each metric space has a completion. One construction takes the Cauchy sequences of $X$ and identifies two of them when the distance between their terms tends to $0$.
The completion is unique: for two completions $(\hat{X}, \iota)$ and $(\hat{X}', \iota')$, there is a unique isometric isomorphism $\varphi \colon \hat{X} \to \hat{X}'$ with $\varphi \circ \iota = \iota'$.
The completion of $\mathbb{Q}$ with the usual metric is $\mathbb{R}$. This is the [[construction of the reals from Cauchy sequences|0D8HE681SC]].
If $X$ is already complete, then $X$ with the identity map is a completion of $X$.

## Explanations

### L1 human

The completion fills the holes of a metric space with new points, and it adds no more points than it needs.
Each new point is a limit of a Cauchy sequence of old points.