A two-point set must be zero-dimensional
HTML articles powered by AMS MathViewer
- by John Kulesza PDF
- Proc. Amer. Math. Soc. 116 (1992), 551-553 Request permission
Abstract:
We prove that any subset of ${\mathbb {R}^2}$ that intersects no line in more than two points either is zero-dimensional or contains an arc. This is used to prove that any subset of ${\mathbb {R}^2}$ that intersects each line in exactly two points is zero-dimensional, answering a question of Mauldin.References
- Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
- Jan van Mill and George M. Reed (eds.), Open problems in topology, North-Holland Publishing Co., Amsterdam, 1990. MR 1078636 S. Mazurkiewicz, Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs, C. R. Varsovie 7 (1914), 382-384.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 551-553
- MSC: Primary 54F45; Secondary 54B05, 54H05
- DOI: https://doi.org/10.1090/S0002-9939-1992-1093599-1
- MathSciNet review: 1093599