Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-23T08:59:48.315Z Has data issue: false hasContentIssue false
  • English
  • Français

On Polish groups admitting a compatible complete left-invariant metric

Published online by Cambridge University Press:  12 March 2014

Maciej Malicki
Maciej Malicki*
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956, Warsaw, Poland, E-mail: mamalicki@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the set of all Polish groups admitting a compatible complete left-invariant metric (called CLI) is coanalytic non-Borel as a subset of a standard Borel space of all Polish groups. As an application of this result, we show that there does not exist a weakly universal CLI group. This, in particular, answers in the negative a question of H.Becker.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 437 - 447
Copyright
Copyright © Association for Symbolic Logic 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Becker, H., Polish group actions: dichotomies and generalized elementary embeddings, Journal of the American Mathematical Society, vol. 11 (1998), no. 2, pp. 397–449.CrossRefGoogle Scholar
[2]Bourbaki, N., Elements of mathematics. General topology. Part 1, Hermann, Paris, 1966.Google Scholar
[3]Christensen, J. P. R., Topology and Borel structure, Mathematical Notes, No. 51, North-Holland Publishing Co., Amsterdam, 1974.Google Scholar
[4]Dobrowolski, T. and Marciszewski, W., Covering a Polish group by translates of a nowhere dense set, Topology and its Applications, vol. 155 (2008), no. 11, pp. 1221–1226.CrossRefGoogle Scholar
[5]Gao, S., Invariant descriptive set theory, CRC Press, 2008.CrossRefGoogle Scholar
[6]Kechris, A., Classical descriptive set theory, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
[7]Malicki, M., An example of a Polish group, this Journal, vol. 73 (2008), no. 4, pp. 1173–1178.Google Scholar
[8]Thompson, A., A metamathematical condition equivalent to the existence of a complete left invariant metric for a Polish group, this Journal, vol. 71 (2006), no. 4, pp. 1108–1124.Google Scholar