View source for Completion

{{MSC|54E50}}
{{TEX|done}}

''of a metric space $(X,d)$''

Given a [[Metric space|metric space]] $(X,d)$, a completion of $X$ is a triple $(Y,\rho,i)$ such that:
*$(Y, \rho)$ is a [[Complete metric space|complete metric space]];
*$i:  X \to Y$ is an [[Isometric mapping|isometric embedding]], namely a map  such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in  X$;
*$i(X)$ is dense in $Y$.
Often people refer to the  metric space $(Y, \rho)$ as the completion. Both the space and the  isometric embedding are unique up to isometries.

The standard construction of the completion is through [[Cauchy sequence|Cauchy sequences]] and can be described as follows. Consider the set $Z$ of all possible Cauchy sequences $\{x_k\}$ of $X$ and introduce the equivalence relation
\[
\{x_k\} \sim \{y_k\} \quad \iff \quad \lim_{k\to\infty} d (x_k, y_k)= 0\, .
\]
$Y$ is then the quotient space $Z/\sim$ endowed with the metric
\[
\rho \left(\left[\{x_k\}\right], \left[\{y_k\}\right]\right) = \lim_{k\to\infty} d (x_k, y_k)\, .
\]
The map $i:X\to Y$ maps each element $x\in X$ in the constant sequence $x_n=x$.

A notion of completion can be introduced in general [[Uniform space|uniform spaces]]: the completion of a metric space is then just a special example. Another notable special example is than the completion of a [[Topological vector space|topological vector space]]. 

The concept of completion was first introduced by Cantor, who defined the space of real numbers as the completion of that of rational numbers, see [[Real number|real number]]


====References====
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|valign="top"|{{Ref|Al}}|| P.S.  Aleksandrov,    "Einführung in die Mengenlehre und die allgemeine   Topologie" ,  Deutsch. Verlag Wissenschaft.  (1984)  (Translated from   Russian)
|-
|valign="top"|{{Ref|Du}}|| J. Dugundji,   "Topology" , Allyn &amp; Bacon  (1966)  {{MR|0193606}} {{ZBL|0144.21501}}
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|valign="top"|{{Ref|Ke}}|| J.L. Kelley,    "General topology" , Springer   (1975) 
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|valign="top"|{{Ref|Ko}}|| G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)
|-
|}