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On Polish groups admitting a compatible complete left-invariant metric

  • Maciej Malicki (a1)

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.

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[1]Becker, H., Polish group actions: dichotomies and generalized elementary embeddings, Journal of the American Mathematical Society, vol. 11 (1998), no. 2, pp. 397449.
[2]Bourbaki, N., Elements of mathematics. General topology. Part 1, Hermann, Paris, 1966.
[3]Christensen, J. P. R., Topology and Borel structure, Mathematical Notes, No. 51, North-Holland Publishing Co., Amsterdam, 1974.
[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. 12211226.
[5]Gao, S., Invariant descriptive set theory, CRC Press, 2008.
[6]Kechris, A., Classical descriptive set theory, Springer-Verlag, New York, 1995.
[7]Malicki, M., An example of a Polish group, this Journal, vol. 73 (2008), no. 4, pp. 11731178.
[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. 11081124.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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