# Definition:Completion (Metric Space)

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## Definition

Let $M_1 = \struct {A, d}$ and $M_2 = \struct {\tilde A, \tilde d}$ be metric spaces.

Then $M_2$ is a **completion** of $M_1$, or $M_2$ **completes** $M_1$, if and only if:

- $(1): \quad M_2$ is a complete metric space
- $(2): \quad A \subseteq \tilde A$
- $(3): \quad A$ is dense in $M_2$
- $(4): \quad \forall x, y \in A : \map {\tilde d} {x, y} = \map d {x, y}$. In terms of restriction of functions, this says that $\map {\tilde d {\restriction_A} } = d$.

It is immediate from this definition that a **completion** of a metric space $M_1$ consists of:

- A complete metric space $M_2$
- An isometry $\phi : A \to \tilde A$

such that $\map \phi A = \set {\map \phi x: x \in A}$ is dense in $M_2$.

An isometry is often required to be bijective, so here one should consider $\phi$ as a mapping from $A$ to the image of $\phi$.

Therefore to insist that $\phi$ be an isometry, *in this context*, is to say that $\phi$ must be an injection that preserves the metric of $M_1$.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces