Two point sets with additional properties

Marek Bienias; Szymon Głąb; Robert Rałowski; Szymon Żeberski

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 4, page 1019-1037
  • ISSN: 0011-4642

Abstract

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A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some σ -ideal, being (completely) nonmeasurable with respect to different σ -ideals, being a κ -covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations of two point sets, namely: partial two point sets and n point sets for n = 3 , 4 , ... , ℵ 0 , ℵ 1 . We obtain consistent results connecting partial two point sets and some combinatorial properties (e.g. being an m.a.d. family).

How to cite

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Bienias, Marek, et al. "Two point sets with additional properties." Czechoslovak Mathematical Journal 63.4 (2013): 1019-1037. <http://eudml.org/doc/260790>.

@article{Bienias2013,
abstract = {A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some $\sigma $-ideal, being (completely) nonmeasurable with respect to different $\sigma $-ideals, being a $\kappa $-covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations of two point sets, namely: partial two point sets and $n$ point sets for $n=3,4,\ldots , \aleph _0,$$\aleph _1.$ We obtain consistent results connecting partial two point sets and some combinatorial properties (e.g. being an m.a.d. family).},
author = {Bienias, Marek, Głąb, Szymon, Rałowski, Robert, Żeberski, Szymon},
journal = {Czechoslovak Mathematical Journal},
keywords = {two point set; partial two point set; complete nonmeasurability; Hamel basis; Marczewski measurable set; Marczewski null; $s$-nonmeasurability; Luzin set; Sierpiński set; two point set; partial two point set; complete nonmeasurability; Hamel basis; Marczewski measurable set; Marczewski null; -nonmeasurability; Luzin set; Sierpiński set},
language = {eng},
number = {4},
pages = {1019-1037},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Two point sets with additional properties},
url = {http://eudml.org/doc/260790},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Bienias, Marek
AU - Głąb, Szymon
AU - Rałowski, Robert
AU - Żeberski, Szymon
TI - Two point sets with additional properties
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 1019
EP - 1037
AB - A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some $\sigma $-ideal, being (completely) nonmeasurable with respect to different $\sigma $-ideals, being a $\kappa $-covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations of two point sets, namely: partial two point sets and $n$ point sets for $n=3,4,\ldots , \aleph _0,$$\aleph _1.$ We obtain consistent results connecting partial two point sets and some combinatorial properties (e.g. being an m.a.d. family).
LA - eng
KW - two point set; partial two point set; complete nonmeasurability; Hamel basis; Marczewski measurable set; Marczewski null; $s$-nonmeasurability; Luzin set; Sierpiński set; two point set; partial two point set; complete nonmeasurability; Hamel basis; Marczewski measurable set; Marczewski null; -nonmeasurability; Luzin set; Sierpiński set
UR - http://eudml.org/doc/260790
ER -

References

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