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Let  be a metric space. Let  be the set of all Cauchy sequences
 in  . Define an equivalence relation  on  by setting
 if the interleave sequence of the sequences  and  is also a Cauchy sequence. The completion of  is defined to be the set  of equivalence
classes of  modulo  .
The metric  on  extends to a metric on  in the following manner:
where  and  are representative Cauchy sequences of elements in  . The definition of  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
 is complete, ensures that the limit exists. The space  with this metric is of course a complete metric space.
The original metric space  is isometric to the subset of  consisting of equivalence classes of constant sequences.
Note the similarity between the construction of  and the construction of
 from
 . The process used here is the same as that used to construct the real numbers
 , except for the minor detail that one can not use the terminology of metric spaces in the construction of
 itself because it is necessary to construct
 in the first place before one can define metric spaces.
If the metric space  has an algebraic structure, then in many cases this algebraic structure carries through unchanged to  simply by applying it one element at a time to sequences in  . We will not attempt to state this principle precisely, but we will mention the following important instances:
- If
 is a topological group, then  is also a topological group with multiplication defined by
- If
 is a topological ring, then addition and multiplication extend to  and make the completion into a topological ring.
- If
 is a field with a valuation  , then the completion of  with respect to the metric imposed by  is a topological field, denoted  and called the completion of  at  .
The completion  of  satisfies the following universal property: for every uniformly continuous map
 of  into a complete metric space  , there exists a unique lifting of  to a continuous map
 making the diagram
commute. Up to isomorphism, the completion of  is the unique metric space satisfying this property. The ability to extend uniformly continuous functions from  to  is often the reason why algebraic structures on  extend to  as described in the previous section.
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"completion" is owned by djao.
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(view preamble)
Cross-references: section, uniformly continuous functions, property, isomorphism, continuous map, lifting, map, uniformly continuous, universal property, topological field, valuation, field, addition, topological ring, multiplication, topological group, algebraic structure, place, necessary, minor, real numbers, similarity, subset, isometric, complete, well defined, limit, metric, equivalence classes, sequences, interleave sequence, equivalence relation, Cauchy sequences, metric space
There are 27 references to this entry.
This is version 5 of completion, born on 2002-05-21, modified 2005-08-21.
Object id is 2923, canonical name is Completion.
Accessed 7090 times total.
Classification:
| AMS MSC: | 13J10 (Commutative rings and algebras :: Topological rings and modules :: Complete rings, completion) | | | 54D35 (General topology :: Fairly general properties :: Extensions of spaces ) | | | 26-00 (Real functions :: General reference works ) |
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Pending Errata and Addenda
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