Polish modules over subrings of $\mathbb Q$

TL;DR

This paper introduces a method to construct Polish modules over subrings of rationals, producing examples of vector spaces with specific embedding properties that answer a question by Frisch and Shinko.

Contribution

The authors develop a novel construction technique for Polish modules over subrings of $Q$, enabling the creation of incomparable Polish $Q$-vector spaces with respect to embeddings.

Findings

Constructed two Polish $Q$-vector spaces $U$ and $V$ with specific embedding properties.

Demonstrated the existence of many incomparable Polish $Q$-vector spaces.

Answered a question posed by Frisch and Shinko regarding embeddings of Polish modules.

Abstract

We give a method of producing a Polish module over an arbitrary subring of $\mathbb Q$ from an ideal of subsets of $\mathbb N$ and a sequence in $\mathbb N$. The method allows us to construct two Polish $\mathbb Q$-vector spaces, $U$ and $V$, such that -- both $U$ and $V$ embed into $\mathbb R$ but -- $U$ does not embed into $V$ and $V$ does not embed into $U$, where by an embedding we understand a continuous $\mathbb Q$-linear injection. This construction answers a question of Frisch and Shinko. In fact, our method produces a large number of incomparable with respect to embeddings Polish $\mathbb Q$-vector spaces.