Metric space completion

Let $(X, \rho)$ is a metric-space.

$(Y, d)$ is a completion of $(X, \rho)$ if $\exists Z$ which is dense-subset of $Y$ and $\exists \varphi$ - metric-space-isometry between $(X, \rho)$ and $(Y, d)$.

Properties

• If $(X, \rho)$ is not complete-metric-space and is dense-subset of $(Y, d)$ which is a complete-metric-space then $(Y, d)$ is a completion of $(X, \rho)$.