On closed embeddings in $P^N \cup Q^N$

Abstract

We prove that if a separable metrizable $X$ is a union of two disjoint 0-dimensional sets $E$, $F$, $E$ is absolutely $G_{\delta}$ and $F$ is absolutely $F_{\sigma\delta}$ 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 $F_{\sigma \delta}$ 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 $H$.