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<p>We give a method of producing a Polish module over an arbitrary subring of ℚ from an ideal of subsets of ℕ and a sequence in ℕ. The method allows us to construct two Polish ℚ-vector spaces, <i>U</i> and <i>V</i>, such that<UnorderedList Mark="Dash"> <ItemContent> <p>both <i>U</i> and <i>V</i> embed into ℝ but</p> </ItemContent> <ItemContent> <p><i>U</i> does not embed into <i>V</i> and <i>V</i> does not embed into <i>U</i>,</p> </ItemContent> </UnorderedList></p><p>where by an embedding we understand a continuous ℚ-linear injection. This construction answers a question of Frisch and Shinko [4]. In fact, our method produces a large number of Polish ℚ-vector spaces which are incomparable with respect to embeddings.</p>

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Polish modules over subrings of ℚ

  • 期刊论文

  • 发表日期：

    2025年11月4日

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  • Dexuan Hu,
  • Sławomir Solecki

摘要

We give a method of producing a Polish module over an arbitrary subring of ℚ from an ideal of subsets of ℕ and a sequence in ℕ. The method allows us to construct two Polish ℚ-vector spaces, U and V, such that

both U and V embed into ℝ but

U does not embed into V and V does not embed into U,

where by an embedding we understand a continuous ℚ-linear injection. This construction answers a question of Frisch and Shinko [4]. In fact, our method produces a large number of Polish ℚ-vector spaces which are incomparable with respect to embeddings.