Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$F_\sigma$-additive families and the invariance of Borel classes


Author: Jirí Spurny
Journal: Proc. Amer. Math. Soc. 133 (2005), 905-915
MSC (2000): Primary 54H05, 54E40; Secondary 28A05
DOI: https://doi.org/10.1090/S0002-9939-04-07587-2
Published electronically: September 20, 2004
MathSciNet review: 2113943
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that any $F_\sigma$-additive family $\mathcal{A}$ of sets in an absolutely Souslin metric space has a $\sigma$-discrete refinement provided every partial selector set for $\mathcal{A}$ is $\sigma$-discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps $F_\sigma$-sets to $F_\sigma$-sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves $F_\sigma$-sets, is shown as an application of the previous result.


References [Enhancements On Off] (What's this?)

  • 1. A.G. El'kin, A-sets in complete metric spaces, Dokl. Akad. Nauk SSSR 175 (1967), 517-520. MR 0214029 (35:4881)
  • 2. S. Graf, Selected results on measurable selections, Rend. Circ. Math. Palermo (1982), 87-122. MR 0683772 (84m:28019)
  • 3. R.W. Hansell, Borel measurable mappings for nonseparable metric spaces, Trans. Amer. Math. Soc. 161 (1971), 145-169. MR 0288228 (44:5426)
  • 4. R.W. Hansell, $F_\sigma$-set covers of analytic spaces and first class selectors, Proc. Amer. Math. Soc. (2) 96 (1986), 365-371. MR 0818473 (87g:54084)
  • 5. R.W. Hansell, Nonseparable analytic metric spaces and quotient maps, Topology Appl. 85 (1998), 143-152. MR 1617459 (99b:54058)
  • 6. R.W. Hansell, On characterizing non-separable analytic and extended Borel sets as types of continuous images, Proc. London Math. Soc. 28 (1974), 683-699. MR 0362269 (50:14711)
  • 7. P. Holický and J. Spurný, Perfect images of absolute Souslin and absolute Borel Tychonoff spaces, Topology Appl. 131 (2003), 281-294. MR 1983084 (2004c:54033)
  • 8. J.E. Jayne and C.A. Rogers, Invariance of Borel classes in metric spaces, Math. Ann. 263 (1983), 323-341. MR 0704298 (84g:54048)
  • 9. J. Kaniewski and R. Pol, Borel-measurable selectors for compact-valued mappings in the non-separable case, Bull. Acad. Polon. Sci. 23 (1975), 1043-1050. MR 0410657 (53:14405)
  • 10. G. Koumoullis, Cantor sets in Prohorov spaces, Fund. Math. 74 (1984), 155-161. MR 0774507 (86c:54038)
  • 11. K. Kuratowski, Topology, Academic Press, New York, 1966. MR 0217751 (36:840)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54H05, 54E40, 28A05

Retrieve articles in all journals with MSC (2000): 54H05, 54E40, 28A05


Additional Information

Jirí Spurny
Affiliation: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: spurny@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/S0002-9939-04-07587-2
Keywords: $F_\sigma$--additive family, $\sigma$--discrete refinement, first class selector, Borel classes
Received by editor(s): April 10, 2003
Received by editor(s) in revised form: October 30, 2003
Published electronically: September 20, 2004
Additional Notes: This research was supported in part by the grant GAČR 201/03/0935, GAČR 201/03/D120 and in part by the Research Project MSM 1132 00007 from the Czech Ministry of Education
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society