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 such that $\phi(X) = \left\{ { \phi(x) : x \in X } \right\}$ is dense in $\tilde X$.
 * A complete metric space $(\tilde X, \tilde d)$
 * An isometry $\phi : X \to \tilde X$

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 $X$.