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Vitali sets and Hamel bases that are Marczewski measurable

Arnold MillerStrashimir Popvassilev — 2000

Fundamenta Mathematicae

We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one...

A MAD Q-set

Arnold W. Miller — 2003

Fundamenta Mathematicae

A MAD (maximal almost disjoint) family is an infinite subset of the infinite subsets of ω = 0,1,2,... such that any two elements of intersect in a finite set and every infinite subset of ω meets some element of in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G δ -set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of P ( ω ) = 2 ω .

Universal functions

Paul B. LarsonArnold W. MillerJuris SteprānsWilliam A. R. Weiss — 2014

Fundamenta Mathematicae

A function of two variables F(x,y) is universal if for every function G(x,y) there exist functions h(x) and k(y) such that G(x,y) = F(h(x),k(y)) for all x,y. Sierpiński showed that assuming the Continuum Hypothesis there exists a Borel function F(x,y) which is universal. Assuming Martin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. Here we show that it is consistent that for each α with 2 ≤ α < ω₁ there...

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