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)

 
 

 

Restrictions to continuous functions and Boolean algebras


Author: Ireneusz Recław
Journal: Proc. Amer. Math. Soc. 118 (1993), 791-796
MSC: Primary 26A21; Secondary 04A15, 06E99, 28A20, 54C30
DOI: https://doi.org/10.1090/S0002-9939-1993-1152289-8
MathSciNet review: 1152289
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that every Borel function $ f:\mathbb{R} \to \mathbb{R}$ is continuous on a set $ A \notin \mathcal{J}$ if $ B(\mathbb{R})/\mathcal{J}$ is weakly distributive. We also show that CCC is not sufficient. We investigate some other conditions considering the problem of restrictions to continuous functions.


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

  • [Bl] H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), 113-128. MR 1501216
  • [Br] Jack B. Brown, Variations on Blumberg's Theorem, Real Anal. Exchange 9 (1983), 123-137.
  • [BP] Jack B. Brown and K. Prikry, Variations on Lusin's theorem, Trans. Amer. Math. Soc. 302 (1987), 77-86. MR 887497 (88e:26003)
  • [BRR] L. Bukovsky, I. Recław, and M. Repicky, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology Appl. 41 (1991), 25-40. MR 1129696 (93b:54037)
  • [CM] J. Cichoń and M. Morayne, Universal functions and generalized classes of function, Proc. Amer. Math. Soc. 102 (1988), 83-89. MR 915721 (89c:26003)
  • [CMPS] J. Cichoń, M. Morayne, J. Pawlikowski, and S. Solecki, Decomposing Baire functions, J. Symbolic Logic 56 (1991), 1273-1283. MR 1136456 (92j:04001)
  • [L] N. Lusin, Sur les ensembles toujurs de premiere categorie, Fund. Math. 21 (1933), 114-126.
  • [M] A. W. Miller, Special subsets of the real line, Handbook of Set Theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, Amsterdam, 1984. MR 776624 (86i:54037)
  • [Ma] E. Marczewski (Szpilrajn), Sur une classe de fonctions de W. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1935), 17-24.
  • [Sh] Juichi Shinoda, Some consequences of Martin's Axiom and the continuum hypothesis, Nagoya Math. J. 49 (1973), 117-125. MR 0319754 (47:8296)
  • [S] W. Sierpiński, Sur un probleme concernant les fonctions semi-continues, Fund. Math. 28 (1937), 1-6.
  • [SZ] W. Sierpiński and A. Zygmund, Sur une fonction qui est discontinue tout ensemble de puissance de continu, Fund. Math. 4 (1923), 316-318.
  • [W] Elżbieta Wagner, Sequences of measurable functions, Fund. Math. 112 (1981), 89-102. MR 619485 (82m:28010)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A21, 04A15, 06E99, 28A20, 54C30

Retrieve articles in all journals with MSC: 26A21, 04A15, 06E99, 28A20, 54C30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1152289-8
Keywords: Continuity, weak distributivity, Boolean algebra, Lusin set
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society