Small combinatorial cardinal characteristics and theorems of Egorov and Blumberg. (English) Zbl 1017.26003

Ciesielski, Krzysztof; Pawlikowski, Janusz

Real Anal. Exch. 26(2000-2001), No. 2, 905-911 (2001).

Summary: We 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 \(f_n: \mathbb{R}\to\mathbb{R}\) of uniformly bounded functions there is a set \(P\subset \mathbb{R}\) of cardinality continuum and an infinite \(W\subset\omega\) such that \(\{f_n\upharpoonright P: n\in W\}\) is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if the functions \(f_n\) are measurable or have the Baire property then \(P\) can be chosen as a perfect set.
We also show that \(\text{cof}({\mathcal N})= \omega_1\) implies existence of a magic set and of a function \(f: \mathbb{R}\to \mathbb{R}\) such that \(f\upharpoonright D\) is discontinuous for every \(D\not\in{\mathcal N}\subset{\mathcal M}\).

MSC:

26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21	Classification of real functions; Baire classification of sets and functions
03E17	Cardinal characteristics of the continuum
26A03	Foundations: limits and generalizations, elementary topology of the line
03E35	Consistency and independence results

Keywords:

cofinality; null sets; uniform convergence; Ramsey ultrafilter; Blumberg theorem; uniformly bounded functions; uniformly continuous functions; magic set