# Vitali sets and Hamel bases that are Marczewski measurable

Arnold Miller; Strashimir Popvassilev

Fundamenta Mathematicae (2000)

- Volume: 166, Issue: 3, page 269-279
- ISSN: 0016-2736

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topMiller, Arnold, and Popvassilev, Strashimir. "Vitali sets and Hamel bases that are Marczewski measurable." Fundamenta Mathematicae 166.3 (2000): 269-279. <http://eudml.org/doc/212481>.

@article{Miller2000,

abstract = {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 for the plane by showing that there is no one-to-one additive Borel map from the plane to the reals.},

author = {Miller, Arnold, Popvassilev, Strashimir},

journal = {Fundamenta Mathematicae},

keywords = {Marczewski measurable set; Vitali set; Hamel basis; Polish space; Marczewski null sets},

language = {eng},

number = {3},

pages = {269-279},

title = {Vitali sets and Hamel bases that are Marczewski measurable},

url = {http://eudml.org/doc/212481},

volume = {166},

year = {2000},

}

TY - JOUR

AU - Miller, Arnold

AU - Popvassilev, Strashimir

TI - Vitali sets and Hamel bases that are Marczewski measurable

JO - Fundamenta Mathematicae

PY - 2000

VL - 166

IS - 3

SP - 269

EP - 279

AB - 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 for the plane by showing that there is no one-to-one additive Borel map from the plane to the reals.

LA - eng

KW - Marczewski measurable set; Vitali set; Hamel basis; Polish space; Marczewski null sets

UR - http://eudml.org/doc/212481

ER -

## References

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- [9] J. H.Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), 1-28. Zbl0517.03018
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