On closed embeddings in $P^N \cup Q^N$
Elżbieta Pol, Roman Pol, Mirosława Reńska
TL;DR
The paper investigates closed embeddings of unions of two disjoint zero-dimensional sets, with one piece absolutely $G_\delta$ and the other absolutely $F_{\sigma\delta}$, into the standard zero-dimensional universes $({\mathbb P}^{\mathbb N}\cup{\mathbb Q}^{\mathbb N})$, ensuring the $G_\delta$ part corresponds to the irrationals and the $F_{\sigma\delta}$ part to the rationals. It shows how to realize such spaces as closed subspaces via a constructive embedding that preserves the respective preimages, and extends these ideas to one-dimensional absolutely $F_{\sigma\delta}$-spaces by embedding into $(({\mathbb P}\cup\{0\})^{\mathbb N})\cup {\mathbb Q}^{\mathbb N}$ with a canonical subset $H$ capturing the $F_{\sigma\delta}$-structure. The authors develop a framework combining compact extensions, dimension controls, and established embedding theorems to achieve the main results (Theorems 1.2 and 1.3), and provide comments on the robustness of ${\mathbb Q}^{\mathbb N}$ as a universal host for zero-dimensional $F_{\sigma\delta}$-spaces and potential extensions to higher Borel classes. Overall, the work identifies universal closed embedding targets for specific Borel-regular decompositions in low-dimensional settings, with implications for understanding the structure of zero-dimensional and one-dimensional $F_{\sigma\delta}$-spaces in Hilbert-cube-like products.
Abstract
We prove that if a separable metrizable $X$ is a union of two disjoint 0-dimensional sets $E$, $F$, $E$ is absolutely $G_δ$ and $F$ is absolutely $F_{σδ}$ then there is a closed embedding $h$ into the union of countable products of the irrationals and the rationals with $E$ being the preimage under $h$ of the countable product of the irrationals and $F$ being the preimage under $h$ of the countable product of the rationals. We prove also that for the set $H$ of points $x$ in the Hilbert cube such that for each $k$ there is $l$ with $x(2^k 3^l)=0$, whenever $A$ is an $F_{σδ}$ set in a compact one-dimensional space $X$, there is an embedding $h$ into the union of the countable product of the irrationals with added point $0$, and the countable product of the rationals, such that $A$ is the preimage under $h$ of the set $H$.