## Real Analysis Exchange

### Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg

#### Abstract

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

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

Dates
First available: 27 June 2008

http://projecteuclid.org/euclid.rae/1214571379

Mathematical Reviews number (MathSciNet)
MR1844405

Zentralblatt MATH identifier
1017.26003

#### Citation

Ciesielski, Krzysztof; Pawlikowski, Janusz. Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg. Real Analysis Exchange 26 (2000), no. 2, 905--912. http://projecteuclid.org/euclid.rae/1214571379.

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