Canonical compactification of a metric space

Question

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed).

But is it possible to compactify a metric space M into M* such that M can be isometrically embedded into M*. Moreover is there a canonical one?

Answer 1

A compact metric space is totally bounded. Inasmuch as total boundedness is hereditary, a metric space which is isometrically embeddable in a compact space must be totally bounded.

Conversely, if a metric space is totally bounded, then its completion is totally bounded and (of course) complete; and a totally bounded complete metric space is compact. (This is because, in a totally bounded metric space, every sequence has a Cauchy subsequence.)

Answer 2

There are examples where a metric space can be isometrically embedded in one of its compactifications. For this to be possible at the very least the metric must be bounded.

For example, take any bounded subset of $\mathbb{R}$; such a subset can be isometrically embedded in its closure, which is a compactification. If the space can be isometrically embedded as a bounded subset of $\mathbb{R}^n$ for some $n$, or as a subset of some cube in $\mathbb{R}^{\omega}$, this same construction will work.