The Metric Completion of a Metric Space

Let $(X,d)$ be a metric space. Let $\mathcal{C}(X)$ be the set of Cauchy sequences in $(X,d)$, $\bar{a}$ the constant sequence with value $a\in X$, $\delta$ the limit distance on $\mathcal{C}(X)$, and $[x]$ the class of $x\in\mathcal{C}(X)$.

1. (Completion)¶ The metric completion of $(X,d)$ is the set $\widehat{X}=\{[x]:\ x\in\mathcal{C}(X)\}$ together with the map $\widehat{d}:\widehat{X}\times\widehat{X}\to\mathbb{R}$ given by

which does not depend on the chosen representatives by Cauchy Sequences in a Metric Space: Constant Sequences, Convergence of Termwise Distances, the Null Relation, and Independence of Representatives §null and Cauchy Sequences in a Metric Space: Constant Sequences, Convergence of Termwise Distances, the Null Relation, and Independence of Representatives §representatives.

2. (Canonical map)¶ The canonical map $\kappa_{X}:X\to\widehat{X}$ is given by $\kappa_{X}(a)=[\bar{a}]$, the class of a Cauchy sequence by Cauchy Sequences in a Metric Space: Constant Sequences, Convergence of Termwise Distances, the Null Relation, and Independence of Representatives §constant.