Geodesic
Two point sets with additional properties
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1019-1037
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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 $\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).
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).
DOI : 10.1007/s10587-013-0069-2
Classification : 03E35, 03E75, 15A03, 28A05
Keywords: two point set; partial two point set; complete nonmeasurability; Hamel basis; Marczewski measurable set; Marczewski null; $s$-nonmeasurability; Luzin set; Sierpiński set 
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     title = {Two point sets with additional properties},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1019--1037},
     year = {2013},
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     language = {en},
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Bienias, Marek; Głąb, Szymon; Rałowski, Robert; Żeberski, Szymon. Two point sets with additional properties. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1019-1037. doi: 10.1007/s10587-013-0069-2

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