The σ-ideal of closed smooth sets does not have the covering property

Carlos Uzcátegui

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 3, page 227-236
  • ISSN: 0016-2736

Abstract

top
We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.

How to cite

top

Uzcátegui, Carlos. "The σ-ideal of closed smooth sets does not have the covering property." Fundamenta Mathematicae 150.3 (1996): 227-236. <http://eudml.org/doc/212173>.

@article{Uzcátegui1996,
abstract = {We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.},
author = {Uzcátegui, Carlos},
journal = {Fundamenta Mathematicae},
keywords = {closed smooth sets; transversals; -ideal of compact sets; Polish space; covering property; analytic set; Borel sets; smooth Borel equivalence relation; Borel basis},
language = {eng},
number = {3},
pages = {227-236},
title = {The σ-ideal of closed smooth sets does not have the covering property},
url = {http://eudml.org/doc/212173},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Uzcátegui, Carlos
TI - The σ-ideal of closed smooth sets does not have the covering property
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 3
SP - 227
EP - 236
AB - We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.
LA - eng
KW - closed smooth sets; transversals; -ideal of compact sets; Polish space; covering property; analytic set; Borel sets; smooth Borel equivalence relation; Borel basis
UR - http://eudml.org/doc/212173
ER -

References

top
  1. [1] J. Burgess, A selection theorem for group actions, Pacific J. Math. 80 (1979), 333-336. Zbl0381.54009
  2. [2] G. Debs et J. Saint Raymond, Ensembles d'unicité et d'unicité au sens large, Ann. Inst. Fourier (Grenoble), 37(3) (1987), 217-239. Zbl0618.42004
  3. [3] L. A. Harrington, A. S. Kechris and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), 903-928. Zbl0778.28011
  4. [4] A. S. Kechris, The descriptive set theory of σ-ideals of compact sets, in: Logic Colloquium'88, R. Ferro, C. Bonotto, S. Valentini and A. Zanardo (eds.), North-Holland, 1989, 117-138. 
  5. [5] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995. 
  6. [6] A. S. Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser. 128, Cambridge Univ. Press, 1987. Zbl0642.42014
  7. [7] A. S. Kechris, A. Louveau and W. H. Woodin, The structure of σ-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 263-288. Zbl0633.03043
  8. [8] A. Louveau, σ-idéaux engendrés par des ensembles fermés et théorèmes d'approximation, Trans. Amer. Math. Soc. 257 (1980), 143-169. Zbl0427.03039
  9. [9] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980. 
  10. [10] S. Solecki, Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), 1022-1031. Zbl0808.03031
  11. [11] C. Uzcátegui, The covering property for σ-ideals of compact sets, Fund. Math. 140 (1992), 119-146. Zbl0799.04008
  12. [12] C. Uzcátegui, Smooth sets for a Borel equivalence relation, Trans. Amer. Math. Soc. 347 (1995), 2025-2039. Zbl0826.03020
  13. [13] B. Weiss, Measurable dynamics, in: Contemp. Math. 26, Amer. Math. Soc., 1984, 395-421. Zbl0599.28023

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.