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

• Autorzy: J. Cichoń , Michał Morayne
• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

analytic sets; universal functions; Baire function; uniform closure; cardinal number

Kategorie tematyczne

• 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1993
• Tom: 142
• Numer: 3
• Strony: 263-268

Daty

wydano

1993

otrzymano

1992-05-19

poprawiono

1992-10-12

Twórcy

autor

J. Cichoń

• Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

autor

Michał Morayne

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

Bibliografia

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