A factorization of metric spaces

We first prove that for every metrizable space $X$, for every closed subset $F\hphantom{i}$ whose complement is zero-dimensional, the space $X$ can be embedded as a closed subset into a product of the closed subset $F$ and a metrizable zero-dimensional space. Using this theorem, we next show the existence of extensors of metrics and ultrametrics which preserve properties of metrics such as completeness, properness, being an ultrametric, its fractal dimensions, and large scale structures. This result contains some of the author’s previous extension theorems for ultrametrics.