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Mazurkiewicz set; GM-set; double midset property
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.
[1] Kulesza J.: A two-point set must be zerodimensional. Proc. Amer. Math. Soc. 116 (1992), 551-553. MR 1093599
[2] Loveland L.D., Loveland S.M.: Planar sets that line hits twice. Houston J. Math. 23 (1997), 485-497. MR 1690037
[3] Mazurkiewicz S.: Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs. C.R. Soc. de Varsovie 7 (1914), 382-383.
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