Universal Spaces for Separable Metric Spaces of Dimension at Most n

Abstract

A member X of a class of spaces $\mathcal {S}$ is called a universal element of the class, if every member of $\mathcal {S}$ can be embedded in X. The purpose of this chapter is to prove that Nöbeling’s space $N_n^{2n+1}$, which consists of all points of $\mathbb {R}^{2n+1}$ that have at most n rational coordinates, is a universal space for the class of all separable metric spaces of covering dimension at most n. We first need some preliminary results.