Metric Completions

Why

We can always work with complete metric spaces.

The justification is that we can always, given an incomplete metric space, construct a larger metric space which contains a subset isomorphic to the original one.

Result

Let $(A, d)$ be an incomplete metric space. There exists a complete metric space $(B, d')$ with $C \subset B$ such that $(A, d)$ and $(C, d')$ are isometric and the image under the isometry of $C$ is dense in $B$.