That is one way of constructing a completion for a metric space. But in general a space has many different completions. To take a very simple example, the open unit interval (0,1) can be completed by adding two points (0 and 1, obviously). It can also be completed by adding just one point, via the mapping $\displaystyle x\mapsto e^{2\pi ix}$, which wraps the interval round a circle and then only the single point 1 needs to be added.

Any locally compact space has a whole family of compactifications, ranging from the one-point compactification (as described above for the unit interval) to the

Stone–Čech compactification. The corona of the algebra of bounded analytic functions on the disk comes somewhere between those two extremes.