The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It

Let $(X,d)$ be a metric space, let $\mathcal{C}(X)$ be the set of its Cauchy sequences, and let $(\widehat{X},\widehat{d})$, $[x]$ and $\kappa_{X}$ be its metric completion, the classes of $x\in\mathcal{C}(X)$ and the canonical map. Convergence of sequences in a metric space is that of Convergent Sequence in a Metric Space, and $\mathbb{N}$ is the set of natural numbers. Continuity of a map between metric spaces means continuity on the whole domain.

1. (Metric)¶ $\widehat{d}$ is a metric on $\widehat{X}$.

2. (Isometry)¶ $\widehat{d}(\kappa_{X}(a),\kappa_{X}(b))=d(a,b)$ for all $a,b\in X$. In particular $\kappa_{X}$ is injective.

3. (Density)¶ For every $x=(x_{k})_{k\in\mathbb{N}}\in\mathcal{C}(X)$, the sequence $(\kappa_{X}(x_{k}))_{k\in\mathbb{N}}$ converges to $[x]$ in $(\widehat{X},\widehat{d})$. In particular $\kappa_{X}(X)$ is dense in the topological space formed by $\widehat{X}$ and the open subsets of $(\widehat{X},\widehat{d})$, a topology by Metric Open Sets Form a Topology.

4. (Completeness)¶ $(\widehat{X},\widehat{d})$ is complete.

5. (Extension)¶ Let $(Y,e)$ be a complete metric space and let $f:X\to Y$ be a map such that $(f(x_{k}))_{k\in\mathbb{N}}$ is a Cauchy sequence in $(Y,e)$ for every $(x_{k})_{k\in\mathbb{N}}\in\mathcal{C}(X)$. Then there is exactly one map $\widehat{f}:\widehat{X}\to Y$ such that

It satisfies $\widehat{f}\circ\kappa_{X}=f$ and is continuous; and if $f$ is Lipschitz with constant $\Lambda$, then so is $\widehat{f}$.

6. (Uniqueness of continuous extensions)¶ Let $(Y,e)$ be a metric space and let $g,h:\widehat{X}\to Y$ be continuous maps with $g\circ\kappa_{X}=h\circ\kappa_{X}$. Then $g=h$.

7. (Inequalities)¶ Let $g,h:\widehat{X}\to\mathbb{R}$ be continuous for the absolute-value metric on $\mathbb{R}$, with $g(\kappa_{X}(a))\le h(\kappa_{X}(a))$ for every $a\in X$. Then $g(\xi)\le h(\xi)$ for every $\xi\in\widehat{X}$.