Completion 2002-05-21 17:30:51 2005-08-21 04:37:08 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage[all]{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\p}{{\mathfrak{p}}} \newcommand{\m}{{\mathfrak{m}}} \newcommand{\M}{{\mathfrak{M}}} \renewcommand{\P}{{\mathfrak{P}}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\N}{\mathbb{N}} \renewcommand{\H}{\mathcal{H}} \newcommand{\A}{\mathcal{A}} \renewcommand{\c}{\mathcal{C}} \renewcommand{\O}{\mathcal{O}} \newcommand{\D}{\mathcal{D}} \newcommand{\lra}{\longrightarrow} \renewcommand{\div}{\mid} \newcommand{\res}{\operatorname{res}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\id}{\operatorname{id}} \newcommand{\diff}{\operatorname{diff}} \newcommand{\incl}{\operatorname{incl}} \newcommand{\Hom}{\operatorname{Hom}} \renewcommand{\Re}{\operatorname{Re}} \newcommand{\intersect}{\cap} \newcommand{\union}{\cup} \newcommand{\bigintersect}{\bigcap} \newcommand{\bigunion}{\bigcup} \newcommand{\ilim}{\,\underset{\longleftarrow}{\lim}\,} 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 {\em 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. \section{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: \begin{enumerate} \item If $(X,\cdot)$ is a topological group, then $\hat{X}$ is also a topological group with multiplication defined by $$ \{x_n\} \cdot \{y_n\} = \{x_n \cdot y_n\}. $$ \item If $X$ is a topological ring, then addition and multiplication extend to $\hat{X}$ and make the completion into a topological ring. \item 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 $v$. \end{enumerate} \section{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 $$ \xymatrix{ X \ar[rr]^f \ar[dr] & & Y \\ & \hat{X} \ar[ur]_{\hat{f}} } $$ 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.