On an extension of the Nöbeling rational universal space
Abstract
A subspace $X$ of the Hilbert cube $I^{\mathbb N}$ is rational if $X = G \cup S$, where $\dim G = 0$ and $S$ is countable. Nöbeling (1934) proved that this class has a universal element, the Nöbeling space $V = \mathbb P^{\mathbb N} \cup \mathbb Q_f^{\mathbb N}$, where $\mathbb P^{\mathbb N}$ consists of points in $I^{\mathbb N}$ with all coordinates irrational, and $\mathbb Q_f^{\mathbb N}$ consists of points with all coordinates rational and all but finitely many coordinates zero. While Nöbeling’s proof was based on intricate geometric reasonings, we give a reasonably simple proof using a different approach: we show that for every $X$, $G$ and $S$ as above, there is an embedding $e:I^{\mathbb N}\to I^{\mathbb N}$ with $e(G) \subset \mathbb P^{\mathbb N}$ and $e(S) \subset \mathbb Q_f^{\mathbb N}$ provided $G \cap S = \emptyset $. We expand $V$ to $V^*$, adding countably many Cantor sets, and we obtain a similar result where $S$ are $\sigma $-compact zero-dimensional sets ($V^*$ is universal for $1$-dimensional spaces).
