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.
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Last modified September 18, 2001.