Real Analysis Exchange

Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg

Krzysztof Ciesielski and Janusz Pawlikowski

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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 $D\notin\mathcal{N}\cap\mathcal{M}$.

Article information

Real Anal. Exchange, Volume 26, Number 2 (2000), 905-912.

First available in Project Euclid: 27 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 03E35: Consistency and independence results
Secondary: 26A03: Foundations: limits and generalizations, elementary topology of the line 03E17: Cardinal characteristics of the continuum

cofinality null sets uniform convergence Ramsey ultrafilter Blumberg theorem magic set


Ciesielski, Krzysztof; Pawlikowski, Janusz. Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg. Real Anal. Exchange 26 (2000), no. 2, 905--912.

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