Let $(X,d)$ be a metric space. Let $\bar{X}$ be the set of all Cauchy sequences $\{x_n\}_{n \in \N}$ in $X$ . Define an equivalence relation $\sim$ on $\bar{X}$ by setting $\{x_n\} \sim \{y_n\}$ if the interleave sequence of the sequences $\{x_n\}$ and $\{y_n\}$ is also a Cauchy sequence. The completion of $X$ is defined to be the set $\hat{X}$ of equivalence classes of $\bar{X}$ modulo $\sim$ .

The metric $d$ on $X$ extends to a metric on $\hat{X}$ in the following manner: $$ d(\{x_n\},\{y_n\}) := \lim_{n \to \infty} d(x_n,y_n), $$ where $\{x_n\}$ and $\{y_n\}$ are representative Cauchy sequences of elements in $\hat{X}$ . The definition of $\sim$ is tailored so that the limit in the above definition is well defined, and the fact that these sequences are Cauchy, together with the fact that $\R$ is complete, ensures that the limit exists. The space $\hat{X}$ with this metric is of course a complete metric space.

The original metric space $X$ is isometric to the subset of $\hat{X}$ consisting of equivalence classes of constant sequences.

Note the similarity between the construction of $\hat{X}$ and the construction of $\R$ from $\Q$ . The process used here is the same as that used to construct the real numbers $\R$ , except for the minor detail that one can not use the terminology of metric spaces in the construction of $\R$ itself because it is necessary to construct $\R$ in the first place before one can define metric spaces.

Metric spaces with richer structure

If the metric space $X$ has an algebraic structure, then in many cases this algebraic structure carries through unchanged to $\hat{X}$ simply by applying it one element at a time to sequences in $X$ . We will not attempt to state this principle precisely, but we will mention the following important instances:

1. If $(X,\cdot)$ is a topological group, then $\hat{X}$ is also a topological group with multiplication defined by $$ \{x_n\} \cdot \{y_n\} = \{x_n \cdot y_n\}. $$
2. If $X$ is a topological ring, then addition and multiplication extend to $\hat{X}$ and make the completion into a topological ring.
3. If $F$ is a field with a valuation $v$ , then the completion of $F$ with respect to the metric imposed by $v$ is a topological field, denoted $F_v$ and called the completion of $F$ at $v$ .

Universal property of completions

The completion $\hat{X}$ of $X$ satisfies the following universal property: for every uniformly continuous map $f: X \lra Y$ of $X$ into a complete metric space $Y$ , there exists a unique lifting of $f$ to a continuous map $\hat{f}: \hat{X} \lra Y$ making the diagram