Geodesic
On Mazurkiewicz sets
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 817-819.

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A Mazurkiewicz set $M$ is a subset of a plane with the property that each straight line intersects $M$ in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.
Classification : 54B20, 54C99, 54F15, 54G20
Keywords: Mazurkiewicz set; GM-set; double midset property 
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Charatonik, Marta N.; Charatonik, Włodzimierz J. On Mazurkiewicz sets. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 817-819. http://geodesic.mathdoc.fr/item/CMUC_2000__41_4_a14/