Metrizable compactification of ω is unique

For every compact metrizable space X there is a metrizable compactification μ(X) of ω whose remainder μ(X)ω is homeomorphic to X. And this μ(X) is unique up to homeomorphisms leaving every point of X fixed. Also μ has a functorial property in the sense that every continuous map X → Y can be extended (not uniquely) to a continuous map (μ(X) → μ(Y); and if the former is surjective so can be the latter. As an application, a simple argument is provided for the fact that the Cantor set is the universal space for the class of all zero-dimensional compact metrizable spaces.