Universal functions and generalized classes of functions

Authors:
J. Cichoń and M. Morayne

Journal:
Proc. Amer. Math. Soc. **102** (1988), 83-89

MSC:
Primary 26A21,; Secondary 04A15

MathSciNet review:
915721

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Abstract | References | Similar Articles | Additional Information

Abstract: For a class of subsets of a set which is closed under countable unions we consider the families of functions

Using universal functions we show that under certain natural assumptions about there exists a function such that there is no partition of and a family of functions such that .

This is a generalization of some results of this type proved by Novikov and Adian, Keldyš, and Laczkovich for the Baire hierarchy of functions. The universal functions technique we use is different from the methods of these authors.

**[1]**S. I. Adian and P. S. Novikov,*On one semicontinuous function*, Uchen. Zap. MGPI W. I. Lenina**138**(3) (1958), 3-10. (Russian)**[2]**L. Keldyš,*Sur les fonctions premières mesurables B*, Dokl. Akad. Nauk SSSR**4**(1934), 192-197. (Russian and French)**[3]**K. Kuratowski,*Topologie*I, PWN, Warszawa, 1958.**[4]**M. Laczkovich, unpublished preprint.**[5]**Arnold W. Miller,*Some properties of measure and category*, Trans. Amer. Math. Soc.**266**(1981), no. 1, 93–114. MR**613787**, 10.1090/S0002-9947-1981-0613787-2**[6]**Yiannis N. Moschovakis,*Descriptive set theory*, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR**561709****[7]**P. S. Novikov,*Collected papers. Set and function theory. Mathematical logic and algebra*, "Nauka", Moscow, 1979. (Russian)**[8]**W. Sierpiński,*Sur un problème concernant les fonctions semi-continues*, Fund. Math.**28**(1937), 1-6.

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0915721-6

Keywords:
Generalized classes of functions,
universal functions,
Baire classes

Article copyright:
© Copyright 1988
American Mathematical Society