completion (original) (raw)

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 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 ℝin the first place before one can define metric spaces.

1 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 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 multiplication defined by
2. 1. If X is a topological ring, then addition and multiplication extend to X^ and make the completion into a topological ring.
3. 1. If F is a field with a valuation v, then the completion ofF 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