Topology/Completion

The completion of a metric space is a very important construction of a complete metric space from a possibly non-complete metric space. In effect, it is the smallest extension of a metric space to a complete version of it.

Definition: Take equivalence classes of Cauchy sequences in a metric space X such that two Cauchy sequences are equivalent when they converge to each other i. e. their difference approaches a limit 0. This set of equivalence classes is called the closure of the metric space.

Theorem

The completion of a metric space is unique and complete, and forms a metric space under the metric d(x_n,y_n) is equal to the limit of their difference, where x_n and y_n are two Cauchy sequences, and this metric is well-defined. Moreover, the subset of the equivalence classes which converge to an element in the metric space X is homeomorphic to X, and any other complete extension of X must contain its completion.

Example

One way of constructing the Real numbers is by saying that it is the completion of the rational numbers.