An abstract version of Sierpiński's theorem and the algebra generated by A and CA functions

J. Cichoń; Michał Morayne

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 3, page 263-268
  • ISSN: 0016-2736

Abstract

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We give an abstract version of Sierpiński's theorem which says that the closure in the uniform convergence topology of the algebra spanned by the sums of lower and upper semicontinuous functions is the class of all Baire 1 functions. Later we show that a natural generalization of Sierpiński's result for the uniform closure of the space of all sums of A and CA functions is not true. Namely we show that the uniform closure of the space of all sums of A and CA functions is a proper subclass of the space of all functions measurable with respect to the least class containing intersections of analytic and coanalytic sets and which is closed under countable unions (A and CA functions are analogues of lower and upper semicontinuous functions, respectively, when measurability with respect to open sets is replaced by that with respect to analytic sets).

How to cite

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Cichoń, J., and Morayne, Michał. "An abstract version of Sierpiński's theorem and the algebra generated by A and CA functions." Fundamenta Mathematicae 142.3 (1993): 263-268. <http://eudml.org/doc/211986>.

@article{Cichoń1993,
abstract = {We give an abstract version of Sierpiński's theorem which says that the closure in the uniform convergence topology of the algebra spanned by the sums of lower and upper semicontinuous functions is the class of all Baire 1 functions. Later we show that a natural generalization of Sierpiński's result for the uniform closure of the space of all sums of A and CA functions is not true. Namely we show that the uniform closure of the space of all sums of A and CA functions is a proper subclass of the space of all functions measurable with respect to the least class containing intersections of analytic and coanalytic sets and which is closed under countable unions (A and CA functions are analogues of lower and upper semicontinuous functions, respectively, when measurability with respect to open sets is replaced by that with respect to analytic sets).},
author = {Cichoń, J., Morayne, Michał},
journal = {Fundamenta Mathematicae},
keywords = {analytic sets; universal functions; Baire function; uniform closure; cardinal number; Sierpiński’s theorem; uniform convergence topology; coanalytic sets},
language = {eng},
number = {3},
pages = {263-268},
title = {An abstract version of Sierpiński's theorem and the algebra generated by A and CA functions},
url = {http://eudml.org/doc/211986},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Cichoń, J.
AU - Morayne, Michał
TI - An abstract version of Sierpiński's theorem and the algebra generated by A and CA functions
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 3
SP - 263
EP - 268
AB - We give an abstract version of Sierpiński's theorem which says that the closure in the uniform convergence topology of the algebra spanned by the sums of lower and upper semicontinuous functions is the class of all Baire 1 functions. Later we show that a natural generalization of Sierpiński's result for the uniform closure of the space of all sums of A and CA functions is not true. Namely we show that the uniform closure of the space of all sums of A and CA functions is a proper subclass of the space of all functions measurable with respect to the least class containing intersections of analytic and coanalytic sets and which is closed under countable unions (A and CA functions are analogues of lower and upper semicontinuous functions, respectively, when measurability with respect to open sets is replaced by that with respect to analytic sets).
LA - eng
KW - analytic sets; universal functions; Baire function; uniform closure; cardinal number; Sierpiński’s theorem; uniform convergence topology; coanalytic sets
UR - http://eudml.org/doc/211986
ER -

References

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  1. [1] J. Cichoń and M. Morayne, Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), 83-89. Zbl0646.26009
  2. [2] J. Cichoń, M. Morayne, J. Pawlikowski and S. Solecki, Decomposing Baire functions, J. Symbolic Logic 56 (1991), 1273-1283. Zbl0742.04003
  3. [3] F. Hausdorff, Set Theory, Chelsea, New York 1962. 
  4. [4] K. Kuratowski, Topology I, Academic Press, New York 1966. 
  5. [5] K. Kuratowski and A. Mostowski, Set Theory, Stud. Logic Found. Math. 86, North-Holland, Amsterdam 1976. 
  6. [6] M. Morayne, Algebras of Borel measurable functions, Fund. Math. 141 (1992), 229-242. Zbl0812.26004
  7. [7] W. Sierpiński, Démonstration d'un théorème sur les fonctions de première classe, ibid. 2 (1921), 37-40. Zbl48.0276.03

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