Small combinatorial cardinal characteristics and theorems of Egorov and Blumberg

by

Krzysztof Ciesielski, and J. Pawlikowski

Real Anal. Exchange 26(2) (2000-2001), 905-911.

We will show that the following set theoretical assumption

• \continuum=\omega2, the dominating number d equals to \omega1, and there exists an \omega1-generated Ramsey ultrafilter on \omega
(which is consistent with ZFC) implies that for an arbitrary sequence fn:R-->R of uniformly bounded functions there is a subset P of R of cardinality continuum and an infinite subset W of \omega such that {fn|P: n in W} is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions fn are measurable or have the Baire property then P can be chosen as a perfect set.

We will also show that cof(null)=\omega1 implies existence of a magic set and of a function f:R-->R such that f|D is discontinuous for every D which is not simultaneously meager and of measure zero.

Version as printed.