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Marczewski–Burstin representations vs. Bernstein and Dense subsets

Nowik A.

Demonstratio Mathematica

2016

Vol. 49, nr 4

372--378

We find values of superposition of Marczewski–Burstin operation and operation of finding Bernstein sets. Several counterexamples of Marczewski–Burstin systems are given.

Uniform algebras in the cantor and baire spaces

Nowik A., Reardon P.

Journal of Applied Analysis

2008

Vol. 14, nr 2

227-238

For each pointclass Γ ⊆ P(2ω) define U[Γ] as the collection of all X⊆ 2ω such that the preimage f− 1(X) belongs to Γ for each continuous f : 2ω → 2ω. We study the properties of and possible rela-tionships among the classes U[Γ], where Γ ranges over the σ-algebras (l), (m), the completely Ramsey sets, and the sets with the Baire property. We also prove some results about cardinal coefficients of U[Γ] for the general case of Marczewski-Burstin representable σ-algebras Γ. We finish by posing some unsolved problems.