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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. In fact, our method produces a large number of incomparable with respect to embeddings Polish Q-vector spaces.
This is joint work with Slawomir Solecki.