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2000/2001 Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg
Krzysztof Ciesielski, Janusz Pawlikowski
Real Anal. Exchange 26(2): 905-912 (2000/2001).


We will show that the following set theoretical assumption

$\mathfrak{c}=\omega_2$, the dominating number $\mathfrak {d}$ equals to $\omega_1$, and there exists an $\omega_1$-generated Ramsey ultrafilter on $\omega$

(which is consistent with ZFC) implies that for an arbitrary sequence $f_n\colon\mathbb{R}\to\mathbb{R}$ of uniformly bounded functions there is a set $P\subset\mathbb{R}$ of cardinality continuum and an infinite $W\subset\omega$ such that $\{f_n\restriction P\colon n\in W\}$ is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions $f_n$ are measurable or have the Baire property then $P$ can be chosen as a perfect set. We will also show that cof$(\mathcal{N})=\omega_1$ implies existence of a magic set and of a function $f\colon\mathbb{R}\to\mathbb{r}$ such that $f\restriction D$ is discontinuous for every $D\notin\mathcal{N}\cap\mathcal{M}$.


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Krzysztof Ciesielski. Janusz Pawlikowski. "Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg." Real Anal. Exchange 26 (2) 905 - 912, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 27 June 2008

zbMATH: 1017.26003
MathSciNet: MR1844405

Primary: 03E35 , 26A15
Secondary: 03E17 , 26A03

Keywords: Blumberg theorem , cofinality , magic set , null sets , Ramsey ultrafilter , Uniform convergence

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 2 • 2000/2001
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