Definition:Completion (Metric Space)

Definition

Let $(X, d)$ and $(\tilde X, \tilde d)$ be metric spaces.

Then $(\tilde X, \tilde d)$ is a completion of $(X, d)$, or $(\tilde X, \tilde d)$ completes $(X, d)$, if:

• $\tilde X$ is complete
• $X \subseteq \tilde X$
• $X$ is dense in $\tilde X$
• $\forall x,y \in X : \tilde d (x, y) = d(x, y)$. In terms of restriction of functions, this says that $\displaystyle \tilde d \restriction_X = d$.

It is immediate from this definition that a completion of a space $(X, d)$ consists of

• A complete metric space $(\tilde X, \tilde d)$
• An isometry $\phi : X \to \tilde X$

such that $\phi(X) = \left\{ { \phi(x) : x \in X } \right\}$ is dense in $\tilde X$.

An isometry is often required to be bijective, so here one should consider $\phi$ as a map from $X$ 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 $X$.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$: Complete Metric Spaces