The Completion of a Metric Space

Let (X, d) be a metric space. The goal of these notes is to construct a complete metric space which contains X as a subspace and which is the “smallest” space with respect to these two properties. The resulting space will be denoted by X and will be called the completion of X with respect to d. The hard part is that we have nothing to work with except X itself, and somehow it seems we have to pull a larger space out of thin air. Indeed, the point is that the completion will be constructed using only X and the metric d, and will not rely on knowing that we already have a space “larger” than X. First we describe the completion as a set. For any Cauchy sequence (xn) in X, let [(xn)] denote the set of all Cauchy sequences (yn) in X such that d(xn, yn) → 0 as n → ∞. So, [(xn)] consists of all Cauchy sequences in X whose terms are getting “closer” and “closer” to the terms in (xn) itself. For example, the Cauchy sequence (xn) is in [(xn)] since d(xn, xn) = 0 is the constant zero sequence and so definitely converges to 0. The intuition is the following. Recall that if X is not complete, the Cauchy sequence (xn) does not necessarily converge and a Cauchy sequence (yn) such that d(xn, yn)→ 0 does not necessarily converge either. However, if these Cauchy sequences did converge, the condition d(xn, yn) → 0 would imply that they converged to the same thing. Similarly, if (xn) and (yn) were Cauchy sequences which did converge and had the same limit, then the sequence (yn) would be in the set [(xn]) we have defined above. The point is that [(xn)] consists of all Cauchy sequences which, if they did converge, would converge to the same thing as (xn) if (xn) converged as well. In other words, we are grouping the Cauchy sequences in X according to whether or not they “should” converge to the same thing.