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# 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

$\xymatrix{X\ar[rr]^{f}\ar[dr]&&Y\\ &\hat{X}\ar[ur]_{{\hat{f}}}}$ |

commute. Up to isomorphism, the completion of $X$ is the unique metric space satisfying this property. The ability to extend uniformly continuous functions from $X$ to $\hat{X}$ is often the reason why algebraic structures on $X$ extend to $\hat{X}$ as described in the previous section.

## Mathematics Subject Classification

54D35*no label found*13J10

*no label found*26-00

*no label found*

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