completion

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

The metric $d$ on $X$ extends to a metric on $\widehat{X}$ in thefollowing manner:

$$d(\{x_n\},\{y_n\}):=\underset{n\to \mathrm{\infty }}{lim}d(x_n,y_n),$$

where $\{x_n\}$ and $\{y_n\}$ are representative Cauchy sequences ofelements in $\widehat{X}$. The definition of $\sim$ is tailored so thatthe limit in the above definition is well defined, and the fact that thesesequences are Cauchy, together with the fact that $ℝ$ is complete,ensures that the limit exists. The space $\widehat{X}$ with this metric is of course a complete metric space.

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

Note the similarity between the construction of $\widehat{X}$ and theconstruction of $ℝ$ from $\mathbb{Q}$. The process used here is the same asthat used to construct the real numbers $ℝ$, except for the minordetail that one can not use the terminology of metric spaces in theconstruction of $ℝ$ itself because it is necessary to construct $ℝ$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 manycases this algebraic structure carries through unchanged to $\widehat{X}$simply by applying it one element at a time to sequences in $X$. Wewill not attempt to state this principle precisely, but we willmention the following important instances:

1. If $(X,\cdot )$ is a topological group, then $\widehat{X}$ is also atopological 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 multiplicationextend to $\widehat{X}$ 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 topologicalfield, denoted $F_v$ and called the completion of $F$ at $v$.

2 Universal property of completions

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