Completions and dense subsets

A subset D is dense in X if its closure is all of X. Equivalently, every nonempty open ball in X meets D. This is a local approximation statement, not a claim that D contains most points by number or measure. The rationals are countable and still dense in the real line. Between any two distinct reals there is a rational, so every real can be approximated to every positive tolerance using rational numbers. The irrational numbers are also dense: density alone does not distinguish their very different cardinality.

For a metric space, density lets you select a sequence from D converging to any given x: choose its nth point within 1/n of x. No single finite tolerance establishes density, because a gap could remain below the resolution tested. A finite grid in an interval is not dense in that interval, though a nested sequence of successively finer grids can have a dense union. The distinction matters in approximation: fixing one discretization and allowing arbitrarily refined discretizations are different mathematical operations.

Consider all Cauchy sequences in X. Two are declared equivalent when the distance between their nth terms tends to zero. Reflexivity and symmetry follow from the metric, while transitivity follows by adding distances through a third sequence. One equivalence class becomes one point of the completion. The same missing real number can be approached by many different rational sequences, and this identification prevents the construction from creating a separate point for every method of approximation. Original points are represented by constant sequences.

The distance between two new points is defined as the limit of their termwise distances. That limit exists because the reverse triangle inequality bounds the difference between two such distances by the sum of the two original sequences' tail distances. The resulting real sequence is Cauchy. Equivalent representatives produce the same limit, so the definition does not depend on arbitrary choices. The triangle inequality passes to the limit, and zero distance means exactly that the representatives belong to the same equivalence class.

The constant-sequence embedding preserves every original distance and has dense image. The constructed space is complete: a Cauchy sequence of its ideal points can be approximated by original points with errors tending to zero, producing a single Cauchy sequence in X representing the desired limit. Any two completions of X are related by a unique isometry that agrees with their embeddings of X. The phrase unique here includes the requirement of fixing the original space; unrelated relabelings are not being counted as different completions.

A uniformly continuous map from X into a complete metric space extends uniquely to the completion by sending an approximating Cauchy sequence to the limit of its images. Uniform continuity ensures the images are Cauchy and that equivalent sequences have identical image limits. Ordinary continuity alone need not suffice. The map 1/x on the positive rational numbers is continuous, but cannot extend continuously with a finite real value at the new point zero. A dense domain determines an extension if one exists; existence requires additional control.