# A dichotomy for Polish modules

@article{Frisch2020ADF, title={A dichotomy for Polish modules}, author={Joshua Frisch and Forte Shinko}, journal={arXiv: Logic}, year={2020} }

Let $R$ be a ring equipped with a proper norm. We show that under suitable conditions on $R$, there is a natural basis under continuous linear injection for the set of Polish $R$-modules which are not countably generated. When $R$ is a division ring, this basis can be taken to be a singleton.

## References

SHOWING 1-10 OF 21 REFERENCES

Homomorphism reductions on Polish groups

- MathematicsArch. Math. Log.
- 2018

It is shown that there is a homomorphism reducible to L of the countable power of the group of increasing homeomorphisms of the unit interval such that every every Kσ subgroup of G^\omega $$Gω is homomorphicism reduces to L.

UNIVERSAL SUBGROUPS OF POLISH GROUPS

- MathematicsThe Journal of Symbolic Logic
- 2014

It is shown that for any locally compact Polish group G, the countable power Gω has a universal Kσ subgroup and a universal compactly generated subgroup.

Universal Abelian topological groups

- Mathematics
- 1999

A topological group is said to be universal in a class of topological groups if and if for every group there is a subgroup of that is isomorphic to as a topological group. A group is constructed…

A Glimm-Effros dichotomy for Borel equivalence relations

- Mathematics
- 1990

A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in…

BOREL CHROMATIC NUMBERS

- Mathematics
- 1999

We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable…

Characterizing Complete Erdős Space

- MathematicsCanadian Journal of Mathematics
- 2009

Abstract The space now known as complete Erdős space ${{\mathfrak{E}}_{\text{c}}}$ was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space ${{\ell }^{2}}$ consisting of all…

Invariant Descriptive Set Theory

- Mathematics
- 2008

Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets…

The graph-theoretic approach to descriptive set theory

- MathematicsBull. Symb. Log.
- 2012

We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.