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Eberly College of Arts and Sciences




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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Mathematics Commons



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