Completion of a metric space

Definition

Let $(X, d)$ be a metric space. A completion of $X$ is a pair $(\hat{X}, \iota)$ of a complete metric space $\hat{X}$ and an isometry $\iota \colon X \to \hat{X}$ with dense 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. 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.