Let (X,d) be a metric space. Let X¯ be the set of all Cauchy sequencesMathworldPlanetmathPlanetmath {xn}n in X. Define an equivalence relationMathworldPlanetmath on X¯ by setting {xn}{yn} if the interleave sequence of the sequencesMathworldPlanetmathPlanetmath {xn} and {yn} is also a Cauchy sequence. The completion of X is defined to be the set X^ of equivalence classesMathworldPlanetmath of X¯ modulo .

The metric d on X extends to a metric on X^ in the following manner:


where {xn} and {yn} are representative Cauchy sequences of elements in X^. 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 completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, ensures that the limit exists. The space X^ with this metric is of course a complete metric space.

The original metric space X is isometric to the subset of X^ consisting of equivalence classes of constant sequences.

Note the similarity between the construction of X^ 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.

1 Metric spaces with richer structure

If the metric space X has an algebraic structurePlanetmathPlanetmath, then in many cases this algebraic structure carries through unchanged to 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. 1.

    If (X,) is a topological group, then X^ is also a topological group with multiplicationPlanetmathPlanetmath defined by

  2. 2.

    If X is a topological ring, then addition and multiplication extend to X^ and make the completion into a topological ring.

  3. 3.

    If F is a field with a valuationPlanetmathPlanetmath v, then the completion of F with respect to the metric imposed by v is a topological field, denoted Fv and called the completion of F at v.

2 Universal property of completions

The completion X^ of X satisfies the following universal propertyMathworldPlanetmath: for every uniformly continuous map f:XY of X into a complete metric space Y, there exists a unique lifting of f to a continuous mapMathworldPlanetmath f^:X^Y making the diagram