Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Classes of Baire functions


Authors: Gregory V. Cox and Paul D. Humke
Journal: Trans. Amer. Math. Soc. 269 (1982), 627-635
MSC: Primary 26A21
DOI: https://doi.org/10.1090/S0002-9947-1982-0637714-8
MathSciNet review: 637714
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{A}$ and $ \mathcal{P}$ denote the sets of approximately continuous and almost everywhere continuous functions, and $ {B_1}(F)$ denote Baire's first class generated by $ F$. The classes $ {B_1}(\mathcal{A})$, $ {B_1}(\mathcal{P})$, $ {B_1}(\mathcal{A} \cap \mathcal{P})$, and Grande's class $ \mathcal{A}{\mathcal{P}_1}$ are investigated in some detail. Although Grande's question of whether $ {B_1}(\mathcal{A} \cap \mathcal{P}) = {B_1}(\mathcal{A}) \cap {B_1}(\mathcal{A}) \cap \mathcal{A}{\mathcal{P}_1}$ is not settled, we do show, among other results, that $ \mathcal{A}{\mathcal{P}_1} \subset {B_1}(\mathcal{P})$.


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

  • [CH] G. Cox and P. Humke, A note on pointwise limits of functions which are approximately continuous and almost everywhere continuous, Studia Sci. Math. Hungar. (to appear). MR 685898 (84g:26002)
  • [GW] C. Goffman and D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116-121. MR 0120327 (22:11082)
  • [G] Z. Grande, Sur les suites de fonctions approximativement continues et continues presque partout, Colloq. Math. 38 (1978), 259-262. MR 0580932 (58:28342)
  • [K] K. Kuratowski, Topology. I, Academic Press, New York, 1966 and PWN, Warsaw, 1958. MR 0217751 (36:840)
  • [M] D. Mauldin, $ \sigma $-ideals and related Baire systems, Fund. Math. 71 (1971), 171-177. MR 0293027 (45:2107)
  • [N] T. Nishiura, The topology of almost everywhere continuous, approximately continuous functions, Acta Math. (to appear).
  • [O$ _{1}$] R. J. O'Malley, Approximately differentiable functions: the $ r$ topology, Pacific J. Math. 72 (1977), 207-222. MR 0447499 (56:5810)
  • [O$ _{2}$] -, Approximately continuous functions which are continuous almost everywhere, Acta Math. Acad. Sci. Hungar. 33 (1979), 395-402. MR 542489 (80g:26006)
  • [P] D. Preiss, Limits of approximately continuous functions, Czechoslovak Math. J. 96 (1971), 371-372. MR 0286947 (44:4154)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A21

Retrieve articles in all journals with MSC: 26A21


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0637714-8
Keywords: Approximately continuous, almost everywhere continuous, Baire function
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society