# Ramsey, Lebesgue, and Marczewski sets and the Baire property

Fundamenta Mathematicae (1996)

- Volume: 149, Issue: 3, page 191-203
- ISSN: 0016-2736

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topReardon, Patrick. "Ramsey, Lebesgue, and Marczewski sets and the Baire property." Fundamenta Mathematicae 149.3 (1996): 191-203. <http://eudml.org/doc/212118>.

@article{Reardon1996,

abstract = {We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.
THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.
THEOREM. In the Ellentuck topology on $[ω]^ω$, $(s)_0$ is a proper subset of the hereditary ideal associated with (s).
We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not $B_r$-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.},

author = {Reardon, Patrick},

journal = {Fundamenta Mathematicae},

keywords = {Ramsey set; Marczewski set; perfect set; measurable set; Baire property; density topology; Ellentuck topology; σ-algebra; perfect sets},

language = {eng},

number = {3},

pages = {191-203},

title = {Ramsey, Lebesgue, and Marczewski sets and the Baire property},

url = {http://eudml.org/doc/212118},

volume = {149},

year = {1996},

}

TY - JOUR

AU - Reardon, Patrick

TI - Ramsey, Lebesgue, and Marczewski sets and the Baire property

JO - Fundamenta Mathematicae

PY - 1996

VL - 149

IS - 3

SP - 191

EP - 203

AB - We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.
THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.
THEOREM. In the Ellentuck topology on $[ω]^ω$, $(s)_0$ is a proper subset of the hereditary ideal associated with (s).
We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not $B_r$-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.

LA - eng

KW - Ramsey set; Marczewski set; perfect set; measurable set; Baire property; density topology; Ellentuck topology; σ-algebra; perfect sets

UR - http://eudml.org/doc/212118

ER -

## References

top- [Br] J. B. Brown, The Ramsey sets and related sigma algebras and ideals, Fund. Math. 136 (1990), 179-185. Zbl0737.28004
- [BrCo]J. B. Brown and G. V. Cox, Classical theory of totally imperfect spaces, Real Anal. Exchange 7 (1982), 1-39.
- [Bu] C. Burstin, Eigenschaften messbaren und nichtmessbaren Mengen, Wien Ber. 123 (1914), 1525-1551.
- [C] P. Corazza, Ramsey sets, the Ramsey ideal, and other classes over ℝ, J. Symbolic Logic 57 (1992), 1441-1468. Zbl0765.03021
- [E] E. Ellentuck, A new proof that analytic sets are Ramsey, ibid. 39 (1974), 163-165. Zbl0292.02054
- [GP] F. Galvin and K. Prikry, Borel sets and Ramsey's theorem, ibid. 38 (1973), 193-198. Zbl0276.04003
- [GW] C. Goffman and D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116-121. Zbl0096.17103
- [GNN] C. Goffman, C. Neugebauer and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497-505. Zbl0101.15502
- [K] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
- [L] A. Louveau, Une démonstration topologique de théorèmes de Silver et Mathias, Bull. Sci. Math. (2) 98 (1974), 97-102. Zbl0311.54043
- [M] E. Marczewski (Szpilrajn), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1935), 17-34. Zbl61.0229.01
- [O] J. C. Oxtoby, Measure and Category, Springer, Amsterdam, 1971. Zbl0217.09201
- [P] S. Plewik, On completely Ramsey sets, Fund. Math. 127 (1986), 127-132. Zbl0632.04005
- [Sc] S. Scheinberg, Topologies which generate a complete measure algebra, Adv. in Math. 7 (1971), 231-239. Zbl0227.28009
- [Si] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60-64. Zbl0216.01304
- [T] F. Tall, The density topology, Pacific J. Math. 62 (1976), 275-284. Zbl0305.54039
- [W] J. T. Walsh, Marczewski sets, measure, and the Baire property. II, Proc. Amer. Math. Soc. 106 (4) (1989), 1027-1030. Zbl0671.28002

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