completion

Let $(X,d)$ be a metric space. Let $\bar{X}$ be the set of all Cauchy sequences $\{x_{n}\}_{n\in\mathbb{N}}$ 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 $\mathbb{R}$ 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 $\mathbb{R}$ from $\mathbb{Q}$ . The process used here is the same as that used to construct the real numbers $\mathbb{R}$ , except for the minor detail that one can not use the terminology of metric spaces in the construction of $\mathbb{R}$ itself because it is necessary to construct $\mathbb{R}$ in the first place before one can define metric spaces.

1 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 $v$ .

2 Universal property of completions

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